1 Introduction
In the classical Construction A, a lattice is obtained by lifting a linear code over some finite ring [18]. This idea was recently generalized to the non-commutative setting by considering natural orders in cyclic algebras over number fields: by taking the quotient of the natural order by a suitable ideal, a ring is obtained which is isomorphic to the quotient of a twisted polynomial ring by some polynomial [19, 41]. This established a connection between twisted polynomials and certain \(\sigma \)-constacyclic codes.
We generalize Construction A using skew polynomial rings \(S[t;\sigma ,\delta ]\) and construct lattices by lifting cyclic \((f,\sigma ,\delta )\)-codes, i.e. much more general linear codes than considered in [19, 41], to lattices in nonassociative algebras. The multiplicative structure of the algebra is not necessary to build a lattice, so we do not limit our considerations to associative algebras as has been done so far.
Anzeige
As recently several classes of cyclic \((f,\sigma ,\delta )\)-codes were constructed with a better minimal distance for certain lengths than previously known codes (e.g., see [5‐11, 15, 20, 27, 36, 59]), \((f,\sigma ,\delta )\)-codes become increasingly important. These codes employ skew polynomial rings \(S[t;\sigma ,\delta ]\) where S is a unital ring, \(\sigma \) an injective endomorphism of S and \(\delta \) a left \(\sigma \)-derivation of S, and are built by choosing a monic polynomial \(f\in S[t;\sigma ,\delta ]\) of degree m, and some monic right divisor g of f [13]. Every cyclic \((f,\sigma ,\delta )\)-code is associated with a principal left ideal of a unital nonassociative algebra \(S_f\) defined by f, which is generated by some monic right divisor g of f.
The nonassociative algebra \(S_f=S[t;\sigma ,\delta ]/S[t;\sigma ,\delta ]f\) is defined on the additive subgroup \(\{h\in S[t;\sigma ,\delta ]\,|\, \mathrm{deg}(h)<m \}\) of \(S[t;\sigma ,\delta ]\) by using right division by f to define the algebra multiplication \(g\circ h=gh \,\,\mathrm{mod}_r f \) [51]. This can be seen as a canonical generalization of associative quotient algebras \(S[t;\sigma ,\delta ]/(f)\), where we factor out a two-sided ideal generated by f, which occurs when Rf is a two-sided ideal. If S is a division algebra, the associative quotient algebras \(S[t;\sigma ,\delta ]/(f)\) as well as the right nuclei of the nonassociative algebras \(S_f\) were used when constructing central simple algebras for instance in [1, 2, 28], [29, Sections 1.5, 1.8, 1.9], [42]. Due to their large nuclei, the algebras \(S_f\) were also successfully employed to systematically build fast-decodable fully diverse space–time block codes in [37, 48, 54], see [49], which are used for reliable high rate transmission over wireless digital channels with multiple antennas transmitting and receiving the data. Skew-polynomial rings and their ideals have been already used in other applications and when generalizing other classical notions like Gröbner bases [3] to a non-commutative setting, e.g. see [14, 16, 31, 32, 34, 35, 44, 45, 58], where they appear as examples of solvable polynomial rings, operator theory [26], and other codes, in particular (cyclic) convolutional codes and MDS codes cf. [21, 22, 24, 25, 38‐40].
We choose suitable monic irreducible skew polynomials \(f\in K[t,\sigma ,\delta ]\) with K/F a finite field extension of number fields, or \(f\in D[t,\sigma ,\delta ]\) with D a cyclic division algebra over a number field, and define natural orders \(\Lambda \) in \(S_f\). We then use the quotient of \(\Lambda \) by certain two-sided ideals to canonically construct a lattice L in \({\mathbb {R}}^N\), i.e. a \({\mathbb {Z}}\)-module L of rank N, from a cyclic \((f,\sigma ,\delta )\)-code over a finite ring.
The non-commutative setup treated in [19, 41] is obtained as the special case where K / F is a cyclic field extension of degree n and \(f(t)=t^n-c\in {\mathcal {O}}_F[t;\sigma ]\) is (right-)invariant, i.e. satisfy \(fR\subset Rf\), which makes Rf a two-sided ideal, and \(S_f\) non-commutative, but still associative.
Anzeige
The advantage of using nonassociative algebras as we do is the fact that this does not limit our choices of skew polynomials f to those which create two-sided ideals Rf. This means that we have a much larger choice of lattices we can build. Lattices now can be obtained by lifting any cyclic \((f,\sigma ,\delta )\)-code, moreover, we can also lift \(\sigma \)-constacyclic codes to lattices (now sitting inside nonassociative algebras). Sometimes there exist easy conditions for nonassociative cyclic algebras to be division algebras which is an additional bonus.
Our Construction A can be used to encode space–time block codes, for coset coding, and in particular for wiretap coding.
The paper is organized as follows: After collecting the results we need in Section 1, for monic and irreducible \(f\in K[t;\sigma ,\delta ]\) we define a natural order in \(S_f\), and investigate the quotients of a natural order by some ideals in Sect. 3. These results are then generalized in Sect. 5 to monic irreducible \(f\in D[t;\sigma ,\delta ]\), where \(D=(K/F,\rho ,c)\) is a cyclic division algebra. In Sects. 4 and 6, we describe a lattice encoding of certain cyclic \((f,\sigma ,\delta )\)-codes over the finite rings \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\), where \({\mathfrak {p}}\) is a maximal ideal in some suitable subring of \({\mathcal {O}}_K\), and how it can be applied to space–time block codes.
2 Preliminaries
2.1 Nonassociative algebras
Let R be a unital commutative ring and let A be an R-module. We call A an algebra over R if there exists an R-bilinear map \(A\times A\rightarrow A\), \((x,y) \mapsto x \cdot y\), denoted simply by juxtaposition xy, the multiplication of A. An algebra A is called unital if there is an element in A, denoted by 1, such that \(1x=x1=x\) for all \(x\in A\). We will only consider unital algebras.
For an R-algebra A, the left nucleus of A is defined as \(\mathrm{Nuc}_l(A) = \{ x \in A \, \vert \, [x, A, A] = 0 \}\) where \([x, y, z] = (xy) z - x (yz)\) for \(x,y,z\in A\), the middle nucleus as \(\mathrm{Nuc}_m(A) = \{ x \in A \, \vert \, [A, x, A] = 0 \}\) and the right nucleus as \(\mathrm{Nuc}_r(A) = \{ x \in A \, \vert \, [A,A, x] = 0 \}\). Their intersection \(\mathrm{Nuc}(A) = \{ x \in A \, \vert \, [x, A, A] = [A, x, A] = [A,A, x] = 0 \}\) is the nucleus of A. The center of A is \(\mathrm{C}(A)=\{x\in A\,|\, x\in \text {Nuc}(A) \text { and }xy=yx \text { for all }y\in A\}\) [53].
Let R be a Noetherian integral domain with quotient field F and A a finite-dimensional unital F-algebra. Then an R-lattice in A is an R-submodule \(\Gamma \) of A which is finitely generated and contains an F-basis of A. An R-order in A is a multiplicatively closed R-lattice containing \(1_A\) (the multiplication may be not associative). An R-order will be called maximal if \(\Gamma '\subset \Gamma \) implies \(\Gamma '=\Gamma \) for every R-order \(\Gamma '\) in A.
An algebra \(A\not =0\) over a field F is called a division algebra, if for any \(a\in A\), \(a\not =0\), the right multiplication with a, \(L_a(x)=ax\), and the right multiplication with a, \(R_a(x)=xa\), are bijective. Any division algebra is simple, that means has only trivial two-sided ideals. A finite-dimensional algebra A is a division algebra over F if and only if A has no zero divisors.
2.2 Skew polynomial rings
Let S be a unital (not necessarily commutative) ring, \(\sigma \) an injective ring homomorphism of S and \(\delta :S\rightarrow S\) a left \(\sigma \)-derivation, i.e. an additive map such that \(\delta (ab)=\sigma (a)\delta (b)+\delta (a)b\) for all \(a,b\in S\), implying \(\delta (1)=0\). Let \(\mathrm{Const}(\delta )=\{a\in S\,|\, \delta (a)=0\}\) and \(\mathrm{Fix}(\sigma )=\{a\in S\,|\, \sigma (a)=a\}\).
The skew polynomial ring \(R=S[t;\sigma ,\delta ]\)(defined first by Ore [43]) is the set of skew polynomials \(a_0+a_1t\) \(+\dots +a_nt^n\) with \(a_i\in S\), where addition is defined term-wise and multiplication by \(ta=\sigma (a)t+\delta (a)\) for all \(a\in S\) (for properties see [17, 23, 26]). The ring \(S[t;\sigma ]=S[t;\sigma ,0]\) is called a twisted polynomial ring and \(S[t;\delta ]=S[t;id,\delta ]\) a differential polynomial ring.
For \(f=a_0+a_1t+\dots +a_nt^n\) with \(a_n\not =0\) define \(\mathrm{deg}(f)=n\) and \(\mathrm{deg}(0)=-\infty \). Then \(\mathrm{deg}(fg)\le \mathrm{deg} (f)+\mathrm{deg}(g)\) with equality ifAn element \(f\in R\) is irreducible in R if it is not a unit and it has no proper factors, i.e if there do not exist \(g,h\in R\) with \(\mathrm{deg}(g),\mathrm{deg} (h)<\mathrm{deg}(f)\) such that \(f=gh\).
-
f has an invertible leading coefficient,
-
g has an invertible leading coefficient,
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S is a domain.
2.3 How to obtain nonassociative algebras from skew polynomial rings
From now on, let \(R=S[t;\sigma ,\delta ]\) and \(\sigma \) injective. We do not assume S to be a division ring. We can still perform a right division by a polynomial \(f \in R\) which has invertible leading coefficient \(d_m\): for all \(g(t)\in R\) of degree \(l> m\), there exist uniquely determined \(r(t),q(t)\in R\) with \(\mathrm{deg}(r)<\mathrm{deg}(f)\), such that \(g(t)=q(t)f(t)+r(t).\) Let \(\mathrm{mod}_r f\) denote the remainder of right division by such an f [51, Proposition 1].
Suppose \(f(t)=\sum _{i=0}^{m}d_it^i\in R=S[t;\sigma ,\delta ]\) has an invertible leading coefficient \(d_m\). Let \(R_m=\{g\in R\,|\, \mathrm{deg}(g)<m\}.\) Then \(R_m\) together with the multiplication \(g\circ h= gh \,\,\mathrm{mod}_r f\) becomes a unital nonassociative ring \(S_f=(R_m,\circ )\) also denoted by R / Rf [51].
This construction was introduced by Petit [46, 47] for unital division rings S. \(S_f\) is a unital nonassociative algebra over \(S_0=\{a\in S\,|\, ah=ha \text { for all } h\in S_f\}\) which is a commutative subring of S. We call \(S_f\) a Petit algebra. The algebra \(S_f\) is associative if and only if Rf is a two-sided ideal in R ([51, Theorem 4 (ii)], or [46, (1)] if S is a division ring). For all invertible \(a\in S\) we have \(S_f\cong S_{af}\), so that without loss of generality it suffices to only consider monic polynomials in the construction.
If \(S_f\) is not associative then \(S\subset \mathrm{Nuc}_l(S_f)\) and \(S\subset \mathrm{Nuc}_m(S_f)\), \(\mathrm{Nuc}_r(S_f)=\{g\in R_m\,|\, fg\in Rf\}\) and \(S_0\) is the center of \(S_f\) [51]. It is easy to see that \(C(S)\cap \mathrm{Fix}(\sigma )\cap \mathrm{Const}(\delta )\subset S_0.\)
If S is a division algebra and \(S_f\) is a finite-dimensional vector space over \(S_0\), then \(S_f\) is a division algebra if and only if f(t) is irreducible in R [46, (9)].
For \(f(t)=\sum _{i=0}^{m}d_it^i\in S[t;\sigma ]\), t is left-invertible in \(S_f\) if and only if \(d_0\) is invertible by a simple degree argument. Thus if f is irreducible (hence \(d_0\not =0\)) and S a division ring then t is always left-invertible in \(S_f\) and \(S_0=\mathrm{Fix}(\sigma )\cap C(S)\) is the center of \(S_f\) [51, Theorem 8 (ii)].
The S-basis \(1, t, t^2, \ldots , t^{n-1}\) is the canonical basis for the left S-module \(S_f\). Since \( S\subset \mathrm{Nuc}_m(S_f)\) and \(S\subset \mathrm{Nuc}_l(S_f)\), the right multiplication with \(0\not =a\in S_f\) in \(S_f\), \(R_h:S_f\longrightarrow S_f,\) \(p\mapsto pa\), is an S-module endomorphism, and after expressing \(R_a\) in matrix form with respect to the canonical basis of \(S_f\), the mapinduces an injective S-linear mapThis fact is exploited when designing space–time block codes which employ one of the following two special cases of algebras:
$$\begin{aligned} \gamma : S_f \rightarrow \mathrm{End}_K(S_f), a\mapsto R_a \end{aligned}$$
$$\begin{aligned} \gamma : S_f \rightarrow \mathrm{Mat}_m(S), a\mapsto R_a \mapsto M(a). \end{aligned}$$
Definition 1
(i)
Let \(S/S_0\) be an extension of commutative unital rings and \(G=\langle \sigma \rangle \) a finite cyclic group of order m acting on S such that \(S_0=\mathrm{Fix}(\sigma )\). For any \(c\in S\), is called a nonassociative cyclic algebra \((S/S_0,\sigma ,c)\) of degree m.
$$\begin{aligned} S_f=S[t;\sigma ]/S[t;\sigma ] (t^m-c) \end{aligned}$$
(ii)
Let D be a finite-dimensional central division algebra over \(F=\mathrm{C}(D)\) of degree n, \(\sigma \in \mathrm{Aut}(D)\) such that \(\sigma |_{F}\) has finite order m and \(f(t)=t^m-d\in D[t;\sigma ]\). Let \(F_0=F\cap \mathrm{Fix}(\sigma )\). The \(F_0\)-algebra \(S_f=D[t;\sigma ]/D[t;\sigma ]f(t)\) is called a (generalized) nonassociative cyclic algebra of degree m. We denote this algebra by \((D,\sigma , d)\) and call \(1,e,\dots ,e^n,t,et\dots ,e^{n-1}t,\dots e^nt^{m-1}\) its canonical basis as a left K-vector space.
Remark 1
If \(c \in S {\setminus } S_0\), then \((S/S_0,\sigma ,c)\) has nucleus S and center \(S_0\). These algebras first appeared over finite fields in [52], over general fields they were studied in [56], and over number fields, in [55]. If \(c\in S_0^\times \), \(S[t;\sigma ]/S[t;\sigma ] (t^m-c)\) is a classical associative cyclic algebra, cf. [19, 41]. If \(c=0\), \(S[t;\sigma ]/S[t;\sigma ] (t^m)\) is a commutative associative algebra, the direct product of m copies of S. If \(S/S_0\) is a cyclic Galois field extension of degree m with Galois group \(\langle \sigma \rangle \) and \(c\in S{\setminus } S_0\), then \(\mathrm{Nuc}((S/S_0,\sigma ,c))=K\). If m is prime then \((S/S_0,\sigma ,c)\) is a division algebra. For non-prime m, a division algebra for all choices of c such that \(1,c,\dots ,c^{m-1}\) are linearly independent [56].
Example 2
Let F and L be fields, \(F_0=F\cap L\), and let K be a cyclic field extension of both F and L such that \(\mathrm{Gal}(K/F) = \langle \rho \rangle \) and \([K:F] = n\), \(\mathrm{Gal}(K/L) = \langle \sigma \rangle \) and \([K:L] = m\), such that \(\rho \) and \(\sigma \) commute. Let \(D=(K/F, \rho , c)\) be an associative cyclic division algebra over F of degree n with canonical basis \(1,e,\dots ,e^{n-1}\) (where \(e^n=c,\) \(el=\rho (l)e\) for every l in K), and \(c\in F_0\). For \(x= x_0 + x_1 e+x_2 e^2+\dots + x_{n-1}e^{n-1}\in D\), extend \(\sigma \) to an automorphism \(\sigma \in \mathrm{Aut}_L(D)\) of order m viaFor all \(d \in D^\times \), \(S_f=D[t;\sigma ]/D[t;\sigma ](t^m-d)\) is the generalized nonassociative cyclic algebra \((D,\sigma ,d)\) of dimension \(m^2n^2\) over \(F_0\). For all \(d \in F^\times \), we have
\((D,\sigma ,d)\) is associative if and only if \(d\in F_0\). For \(f\in F_0[t]\), \((D,\sigma ,d)\) is a generalized cyclic algebra of degree n [29, Section 1.4].
$$\begin{aligned} \sigma (x)=\sigma (x_0) + \sigma (x_1)e +\sigma (x_2) e^2 +\dots +\sigma (x_{n-1}) e^{n-1}. \end{aligned}$$
$$\begin{aligned} S_f=D[t;\sigma ]/D[t;\sigma ](t^m-d)= (L/F_0,\rho , c)\otimes _{F_0} (F/F_0, \sigma , d)=(D, \sigma , d) \end{aligned}$$
2.4 Space–time block coding
An (\(s\times t\)) space–time block code (STBC) is a set \({\mathcal {C}}\) of complex \(s\times t\) matrices. \({\mathcal {C}}\) is called linear if \(X,X'\in {\mathcal {C}}\) implies \(X\pm X'\in {\mathcal {C}}\). A linear code is called fully diverse, if \(\mathrm{det}X\not =0\) for all \(0\not =X\in {\mathcal {C}}\).
Let K/F be a Galois field extension of degree n and K an imaginary number field. Nonassociative cyclic division algebras \(A=(K/F,\sigma , c)\) of degree n can be used to build linear \(n\times n\) STBCs with entries in K, since the right multiplication in A induces the injective K-linear map \(\gamma : A \hookrightarrow \mathrm{End}_K(A)\hookrightarrow \mathrm{Mat}_(K)\), \( a \mapsto R_a\mapsto M(a)\) (cf. Sect. 2.3). The set of matrices \(\gamma (A)\) is a linear STBC that is fully diverse since A is a division algebra.
Let \(A=(D,\sigma ,d)\) be a generalized nonassociative cyclic division algebra, with \(D=(K/F,\sigma , c)\) an associative cyclic algebra of degree n. Again, A can be used to build a fully diverse linear \(mn\times mn\) STBC with entries in K: we know \(\gamma : A \hookrightarrow \mathrm{End}_D(A), a\mapsto R_a\) is an injective D-linear map, and \(K\subset D\). Using the canonical K-basis of A, we obtain an \(mn\times mn\)-matrix M(a) representing \(R_a\) for every \(a\in A\). Thus we have \(\gamma : A \hookrightarrow \mathrm{End}_D(A)\hookrightarrow \mathrm{Mat}_{mn}(K),\) \( a \mapsto R_a\mapsto M(a)\) and \(\gamma (A)\) is a fully diverse linear STBC.
When \(d\in L^\times \) or \(d\in F^\times \), \(\gamma (A)\) is used for the codes in [49, 50, 54]. For \(m=2\), \(\gamma (A)\) is used in the iterated codes constructed in [37]. In particular, for \(d\in F^\times \) the algebra in Example 2 is employed for the space–time block codes in [54], see also [48].
2.5 Cyclic \((f,\sigma ,\delta )\)-codes
Let \(f\in S[t;\sigma ,\delta ]\) be monic of degree m and \(\sigma \) injective. We associate to an element \(a(t)=\sum _{i=0}^{m-1}a_it^i\) in \(S_f\) the vector \((a_0,\dots ,a_{m-1})\). A linear code of length m over S is a submodule of the S-module \(S^m\). Conversely, for any linear code \({\mathcal {C}}\) of length m we denote by \({\mathcal {C}}(t)\) the set of skew polynomials \(a(t)=\sum _{i=0}^{m-1}a_it^i\in S_f\) associated to the codewords \((a_0,\dots ,a_{m-1})\in {\mathcal {C}}\).
A cyclic \((f,\sigma ,\delta )\)-code \({\mathcal {C}}\subset S^m\) is a set consisting of the vectors \((a_0,\dots ,a_{m-1})\) obtained from elements \(h=\sum _{i=0}^{m-1}a_it^i\) in a left principal ideal \(S_f g\) where \(S_f=S[t;\sigma ,\delta ]g/S[t;\sigma ,\delta ]f\), and g is a monic right divisor of f. A code \({\mathcal {C}}\) over S is called \(\sigma \)-constacyclic if there is a non-zero \(c\in S\) such that
$$\begin{aligned} (a_0,\dots ,a_{m-1})\in {\mathcal {C}}\Rightarrow (\sigma (a_{m-1})c,\sigma (a_0),\dots ,\sigma (a_{m-2}))\in {\mathcal {C}}. \end{aligned}$$
Lemma 3
(cf. [51, Proposition 7]) Let \(f\in R=S[t;\sigma ,\delta ]\) be monic of degree m.
(a)
Let \(\sigma \) be injective. Then:
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Every right divisor g of f of degree \(<m\) with an invertible leading coefficient generates a principal left ideal in \(S_f\).
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All left ideals in \(S_f\) which contain a non-zero polynomial g of minimal degree with invertible leading coefficient are principal left ideals, and g is a right divisor of f in R.
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([13, Theorem 1]) Each principal left ideal generated by a monic right divisor of f is an S-module which is isomorphic to a submodule of \(S^m\) and forms a code of length m and dimension \(m-\mathrm{deg}(g)\).
(b)
Let S be a division ring. Then all left ideals in \(S_f\) are generated by some monic right divisor g of f in R.
Proof
(a)
Let g(t) be such a right divisor of f(t), then the ideal Rf is contained in Rg and it is easy to check that \(Rg/Rf=\{h\in R_m\,|\, h=sg \text { for some } s\in R_m\}\) is a left ideal in \(S_f\).
The proof of the second assertion is similar to the one of [12, Lemma 1]: Suppose that I is a left ideal in \(S_f\) which contains a non-zero polynomial g of minimal degree with invertible leading coefficient. For any \(p\in I\subset R_m\), a right division by g yields unique \(r,q\in R\) with \(\mathrm{deg}(r)<\mathrm{deg}(g)\) such that \(p=qg+r \) and hence \(r=p-qg\in I\). Since we chose \(g\in I\) to have minimal degree, we conclude that \(r=0\), implying \(p=qg\) and so \(I=Rg\) is a principal left ideal, and g is a right divisor of f in R.
(b)
Let I be a left ideal of \(S_f\). If \(I=\{0\}\) then \(I=(0)\). So suppose \(I\not =(0)\) and choose a monic non-zero polynomial g in \(I\subset R_m\) of minimal degree. As in the proof of (i), for any \(p\in I\), a right division by g yields unique \(r,q\in R\) with \(\mathrm{deg}(r)<\mathrm{deg}(g)\) such that \(p=qg+r \) and hence \(r=p-qg\in I\). Since \(g\in I\) has minimal degree, \(r=0\), and so \(I=Rg\). \(\square \)
Let \(f,g,h,h'\in S[t;\sigma ,\delta ] \) be monic polynomials such that \(f=gh=h'g\). Let \({\mathcal {C}}\) be the cyclic \((f,\sigma ,\delta )\)-code corresponding to g and \(c(t)=\sum _{i=0}^{m-1}c_it^i\in S[t;\sigma ,\delta ]\). Then \((c_0,\dots ,c_{m-1})\in {\mathcal {C}}\) is equivalent to \(c(t)h(t)=0\) in \(S_f\) [13, Theorem 2], i.e. h is a parity check polynomial for \({\mathcal {C}}\).
The codes \({\mathcal {C}}\) of length m we consider consist of all elements \((a_0,\dots ,a_{m-1})\) obtained from polynomials \(a(t)=\sum _{i=0}^{m-1}a_it^i\) in a left principal ideal \(S_f g\) of \(S_f\), with g a monic right divisor of f; \(\sigma \)-constacyclic codes are obtained when \(f(t)=t^m-c\in S[t;\sigma ]\).
For a field K, every skew polynomial ring \(K[t;\sigma ,\delta ]\) can be made into either a twisted or a differential polynomial ring by a linear change of variables [29, 1.1.21]. When constructing linear codes, however, we will consider general skew polynomial rings. They might produce better distance bounds than cyclic \((f,\sigma ,\delta )\)-codes constructed only with an automorphism, where \(\delta =0\), see [8] for examples of this phenomenon.
3 Natural orders in \(S_f\) and their quotients by a prime ideal, I
In the following, we use the notation from [19, Section 2]. Let K / F be a Galois extension of number fields of degree n with \({\mathcal {O}}_F\) and \({\mathcal {O}}_K\) the rings of integers of F, respectively K.
3.1 The setup
Let \({\mathfrak {p}}\) be a maximal ideal of \({\mathcal {O}}_F\), p the prime lying below \({\mathfrak {p}}\) and \({\mathcal {O}}_F/{\mathfrak {p}}={\mathbb {F}}_{p^j}\), where j is the inertial degree of \({\mathfrak {p}}\) above p. Let \(\pi :{\mathcal {O}}_K\longrightarrow {\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\) be the canonical projection. Let \(\sigma \in G=\mathrm{Gal}(K/F)\). We have \(\sigma ({\mathfrak {p}}{\mathcal {O}}_K)\subset {\mathfrak {p}}{\mathcal {O}}_K\) since \(\sigma |_F=id\). Thus \(\sigma \) induces a ring homomorphismwith \(\pi \circ \sigma |_{{\mathcal {O}}_K}=\overline{\sigma } \circ \pi \) and \(\mathrm{Fix}(\overline{\sigma })={\mathbb {F}}_{p^j}\). Suppose that \(\delta \) is an F-linear left \(\sigma \)-derivation on K such that \(\delta ({\mathcal {O}}_K)\subset {\mathcal {O}}_K\). Then \(\delta \) induces a left \(\overline{\sigma }\)-derivation \(\overline{\delta }:{\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K \longrightarrow {\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K.\) Since \({\mathcal {O}}_K\) is a Dedekind domain we havefor suitable prime (maximal) ideals \({\mathfrak {p}}_i\) of \({\mathcal {O}}_K\), \(e_i\ge 0\). The ideals \({\mathfrak {p}}_t^{e_t}\) are pair-wise comaximal. By the Chinese Remainder Theorem, we have thus the following direct sum of rings:
G acts trivially on each of these \( {\mathfrak {p}}_t\), therefore there is an induced action of G on each \({\mathcal {O}}_K/ {\mathfrak {p}}_t^{e_t}{\mathcal {O}}_K\) and the above is an isomorphism of G-modules (cf. [41, (9)]). That means on each ring \({\mathcal {O}}_K/ {\mathfrak {p}}_t^{e_t}{\mathcal {O}}_K\) there is a canonical induced automorphism \(\overline{\sigma }\) and a canonical left \(\overline{\sigma }\)-derivation \(\overline{\delta }\) induced by \(\delta \).
$$\begin{aligned} \overline{\sigma }:{\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K \longrightarrow {\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K, \quad a+{\mathfrak {p}}{\mathcal {O}}_K \mapsto \sigma (a)+{\mathfrak {p}}{\mathcal {O}}_K \end{aligned}$$
$$\begin{aligned} {\mathfrak {p}}{\mathcal {O}}_K={\mathfrak {p}}_1^{e_1}{\mathfrak {p}}_2^{e_2}\dots {\mathfrak {p}}_g^{e_g} \end{aligned}$$
$$\begin{aligned} {\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K= {\mathcal {O}}_K/ {\mathfrak {p}}_1^{e_1} \cdots {\mathfrak {p}}_g^{e_g}{\mathcal {O}}_K \cong {\mathcal {O}}_K/ {\mathfrak {p}}_1^{e_1}{\mathcal {O}}_K\times \dots \times {\mathcal {O}}_K/ {\mathfrak {p}}_g^{e_g}{\mathcal {O}}_K. \end{aligned}$$
In particular, if \({\mathfrak {p}}\) is inert in K / F, \({\mathfrak {p}}{\mathcal {O}}_K\) is a prime ideal in \({\mathcal {O}}_K\) and thus \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K={\mathbb {F}}_{p^{nj}}\) a finite field, and \( \overline{\sigma }\in \mathrm{Gal}({\mathbb {F}}_{p^{nj}}/{\mathbb {F}}_{p^{j}})\) (cf. [19, Section 2] if \(\delta =0\) and K / F is cyclic).
3.2 Natural orders
Suppose \(f(t)=\sum _{i=0}^{m}d_it^i\in {\mathcal {O}}_K[t;\sigma ,\delta ]\) is monic and irreducible in \(K[t;\sigma ,\delta ]\). Consider the nonassociative division algebraover F. Then the nonassociative \({\mathcal {O}}_{F}\)-algebrais an \({\mathcal {O}}_{F}\)-order in \(S_f\) called the natural order and \(\Lambda ={\mathcal {O}}_K\oplus {\mathcal {O}}_K t\oplus \dots \oplus {\mathcal {O}}_K t^{m-1}\) as left \({\mathcal {O}}_K\)-module. Since f is irreducible in \(K[t;\sigma ,\delta ]\), \(\Lambda \) does not have any zero divisors. The center of \(\Lambda \) contains \({\mathcal {O}}_{F}\). Hence for every maximal ideal \({\mathfrak {p}}\) in \({\mathcal {O}}_{F}\), \({\mathfrak {p}}\Lambda \) is a two-sided ideal of \(\Lambda \).
$$\begin{aligned} S_f=K[t;\sigma ,\delta ]/K[t;\sigma ,\delta ] f \end{aligned}$$
$$\begin{aligned} \Lambda ={\mathcal {O}}_K[t;\sigma ,\delta ]/{\mathcal {O}}_K[t;\sigma , \delta ]f \end{aligned}$$
\(\Lambda \) is usually not maximal, but it is uniquely determined whenever Rf is not a two-sided ideal, since in that case K is the left and middle nucleus of \(S_f\) and uniquely determines \({\mathcal {O}}_{K}\) and in turn \(\Lambda \). (For examples of classes of maximal orders in nonassociative cyclic algebras of degree two, cf. [30, 33], the results there can be generalized to nonassociative algebras of any degree n.)
Remark 4
For any \(g(t)=\sum _{i=0}^{m-1}a_i t^i\in {\mathcal {O}}_K[t;\sigma ,\delta ]\) define \(\overline{g}(t)=\sum _{i=0}^{m-1}\overline{a_i}t^i\in ({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K)[t;\overline{\sigma }, \overline{\delta }]\) with \(\overline{a_i}= a_i+{\mathfrak {p}}{\mathcal {O}}_K\). Let \(\overline{f}(t)=\sum _{i=0}^{m}\overline{d_i}t^i\in ({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K)[t;\overline{\sigma }, \overline{\delta }]\) with \(\overline{d_i}= d_i+{\mathfrak {p}}{\mathcal {O}}_K\).
Lemma 5
(i)
The surjective homomorphism of nonassociative rings has kernel \({\mathfrak {p}}\Lambda \).
$$\begin{aligned} \Psi :\Lambda \longrightarrow S_{\overline{f}},\quad g\mapsto \overline{g} \end{aligned}$$
(ii)
\(\Psi \) induces an \({\mathbb {F}}_{p^j}\)-algebra isomorphism given by
$$\begin{aligned} \Psi :\Lambda /{\mathfrak {p}}\Lambda \longrightarrow S_{\overline{f}},\quad g+{\mathfrak {p}}\Lambda \mapsto \overline{g}. \end{aligned}$$
Proof
(i)
\(\Lambda \) is nonassociative \({\mathcal {O}}_{F}\)-algebra and \(\Psi \) is a well-defined surjective homomorphism with kernel \({\mathfrak {p}}\Lambda \): For all \(g=\sum _{i=0}^{m-1}b_it^i\in {\mathfrak {p}}\Lambda \) it follows that \(\overline{g} =\sum _{i=0}^{m-1}\overline{b_i}t^i=0\) in \(({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K)[t;\overline{\sigma }, \overline{\delta }]\), so that \({\mathfrak {p}}\Lambda \subset ker(\Psi )\).
Suppose conversely there is a nonzero \(g\in \Lambda \) such that \(\Psi (g)=\overline{g}=0\), then \(g=hf+r\) in \({\mathcal {O}}_K[t;\sigma ,\delta ]\) with a nonzero \(r\in {\mathcal {O}}_K[t;\sigma ,\delta ]\), and so \(\overline{g}=\overline{h}\,\overline{f}+\overline{r}\) with \(\overline{r}=0\) in \(({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K)[t;\overline{\sigma }, \overline{\delta }]\). We have \({\mathfrak {p}}\Lambda =\{al\,|\, a\in {\mathfrak {p}},l\in \Lambda \}=\{\sum _{i=0}^{m-1}a_it^i\,|\, a_i\in {\mathfrak {p}}{\mathcal {O}}_K\}.\) This implies that \(r\in {\mathfrak {p}}{\mathcal {O}}_K[t;\sigma ,\delta ]={\mathfrak {p}}\Lambda \).
(ii)
follows from (i). \(\square \)
Example 6
Let \(\mathrm{Gal}(K/F)=\) \(<\sigma>\) and \(f(t)=t^n-d\in {\mathcal {O}}_K[t;\sigma ]\) irreducible in \(K[t;\sigma ]\). \(A=(K/F,\sigma ,d)\) is a nonassociative cyclic division algebra of degree n over F and \(S_{\overline{f}}=\big (({\mathcal {O}}_K/{\mathfrak {p}} {\mathcal {O}}_K)/{\mathbb {F}}_{p^j},\overline{\sigma },\overline{d}\big )\) with \(\overline{f}(t)=t^n-\overline{d}\in ({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K)[t;\overline{\sigma }].\) If \(d\in {\mathcal {O}}_F\) is non-zero, A is associative and \(\Lambda \) depends on the choice of the maximal subfield K in A. Then \(S_{\overline{f}}\) is an associative (generalized) cyclic algebra as in [19, 41] and \(\overline{f}(t)\) is reducible whenever \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\) is a field.
If \(d\in {\mathcal {O}}_K{\setminus } {\mathcal {O}}_F\), A is not associative and the natural orderof A is uniquely determined. If n is prime then f is irreducible and A a division algebra for every \(d\in {\mathcal {O}}_K{\setminus } {\mathcal {O}}_F\). If n is not prime and \(1,d,\dots ,d^{n-1}\) are linearly independent then f is irreducible and A a division algebra (Remark 1). Furthermore,
$$\begin{aligned} \Lambda ={\mathcal {O}}_K[t;\sigma ]/{\mathcal {O}}_K[t;\sigma ] f={\mathcal {O}}_K \oplus {\mathcal {O}}_K t\oplus \dots \oplus {\mathcal {O}}_K t^{n-1} \end{aligned}$$
$$\begin{aligned} \Lambda /{\mathfrak {p}}\Lambda \cong (({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K)/{\mathbb {F}}_{p^j}, \overline{\sigma },\overline{d})=S_{\overline{f}}. \end{aligned}$$
Remark 7
Suppose that K / F is cyclic of degree n and inertial with respect to \({\mathfrak {p}}\), then \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K= {\mathbb {F}}_{p^{jn}}\) and \(\mathrm{Gal}({\mathbb {F}}_{p^{jn}}/{\mathbb {F}}_{p^{j}})=\langle \overline{\sigma } \rangle \). Let \(f(t)=\sum _{i=0}^{n}d_it^i\in {\mathcal {O}}_K[t;\sigma ,\delta ]\) be such that \({\overline{f}}(t)=t^n-\overline{ d_0}\).
is always a division algebra, i.e. \(f(t)=t^n-\overline{d_0}\) is irreducible, and so there are no non-trivial left ideals by Lemma 3 (b).
(i)
In [19], only polynomials \(f(t)=t^n-d_0\) with \(d_0\in {\mathcal {O}}_F\) are considered which makes the ideal \(({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K)[t;\overline{\sigma }] \overline{f}\) two-sided and the resulting \({\mathbb {F}}_{p^j}\)-algebra associative. In this case, \(\overline{f}(t)=t^n-\overline{d_0}\) is always reducible in \({{\mathbb {F}}}_{p^{jn}}[t;\overline{\sigma }]\).
(ii)
By Lemma 3, if \(\overline{f}\) is irreducible, then \(S_{\overline{f}}\) has no non-trivial left ideals. For instance, if n is prime and \(d_0\not \in {\mathcal {O}}_F\) then for all \(\overline{d_0}\not =0\),
$$\begin{aligned} \Lambda /{\mathfrak {p}}\Lambda \cong ({\mathbb {F}}_{p^{jn}}/ {\mathbb {F}}_{p^j},\overline{\sigma },\overline{d_0}) \end{aligned}$$
4 Lattice encoding of cyclic \((f,\sigma ,\delta )\)-codes over \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\), I
We keep the assumptions and notation from Sect. 3. Let \(\mathcal {I}=\Lambda g(t)\) be a principal left ideal of \(\Lambda \) generated by a monic polynomial g(t) such that \({\mathfrak {p}}\subset \mathcal {I}\cap {\mathcal {O}}_F\). Then \(\mathcal {I}/{\mathfrak {p}}\Lambda \) is a principal left ideal of \(\Lambda /{\mathfrak {p}}\Lambda \) and \(\Psi (\mathcal {I}/{\mathfrak {p}}\Lambda )\) is a principal left ideal of \((({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K)/{\mathbb {F}}_{p^j},\overline{\sigma },\overline{c})\) generated by the monic polynomial \(\Psi (g+{\mathfrak {p}}\Lambda )=\overline{ g}\). That means, \(\Psi (\mathcal {I}/{\mathfrak {p}}\Lambda )\) corresponds to an \((\overline{f},\overline{\sigma },\overline{\delta })\)-code \({\mathcal {C}}\) over \({\mathbb {F}}_q\). In particular, if we choose f(t) such that \(\overline{f}(t)=t^m-\overline{ c}\) with \(\overline{ c}\) non-zero, then \(\Psi (\mathcal {I}/{\mathfrak {p}}\Lambda )\) corresponds to a \(\overline{\sigma }\)-constacyclic code over \({\mathbb {F}}_q\).
If \(\overline{f}\) is irreducible and \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\) a field, then \(S_{\overline{f}}\) has no nontrivial principal left ideals which contain a non-zero polynomial of minimal degree with invertible leading coefficient and so \({\mathcal {C}}\) has length n and dimension n, or is zero, whereas when \(\overline{f}\) is reducible and \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\) a field, an \((\overline{f},\overline{\sigma },\overline{\delta })\)-code \({\mathcal {C}}\) corresponds to a right divisor \(\overline{g}\) of \(\overline{f}\) and has dimension \(n-\mathrm{deg}(\overline{g})\). So we will look for irreducible f where \(\overline{f}\) is reducible.
4.1 Construction A
Letbe the canonical projection \(\Lambda \longrightarrow \Lambda /{\mathfrak {p}}\Lambda \) composed with \(\Psi \). We know that \({\mathcal {O}}_K\) is a free \({\mathbb {Z}}\)-module of rank \(n[F:\mathbb {Q}]\). Thenis a \({\mathbb {Z}}\)-module of dimension \(N=nm[F:\mathbb {Q}]\). The embedding of this lattice into \({\mathbb {R}}^N\) is canonically determined by considering \(A\otimes _{\mathbb {Q}}{\mathbb {R}}\). Now all works exactly as as explained in [19, Section 3.3]. The construction of L can be seen as a non-commutative variation of the classical Construction A in [18].
$$\begin{aligned} \rho : \Lambda \longrightarrow \Lambda /{\mathfrak {p}}\Lambda \longrightarrow \Psi (\Lambda /{\mathfrak {p}}\Lambda ) \end{aligned}$$
$$\begin{aligned} L=\rho ^{-1}({\mathcal {C}})=\mathcal {I} \end{aligned}$$
This way we can construct a lattice L in \({\mathbb {R}}^N\) from the linear code \({\mathcal {C}}\) over the finite ring \(S={\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\). The non-commutative variation of Construction A in [19] is the special case that \(f(t)=t^n-c\in {\mathcal {O}}_F[t]\subset K[t;\sigma ]\), where \(S_f\) is associative.
Example 8
Let \(K[t;\sigma ]=\mathbb {Q}(i)[t;\sigma ]\) with \(\sigma \) the complex conjugation, so that \(F=\mathbb {Q}\), \({\mathcal {O}}_F={\mathbb {Z}}\) and \({\mathcal {O}}_K={\mathbb {Z}}[i]\). Let \(f(t)=t^2-t+(i-3)\in {\mathbb {Z}}[i][t,\sigma ]\), then f(t) is irreducible in \(\mathbb {Q}(i)[t;\sigma ]\), since \(\sigma (z)z-z\not =i-3\) for all \(z\in \mathbb {Q}(i)\) [46, (17)]. Let \(p=3\). Then \({\mathbb {Z}}[i]/3{\mathbb {Z}}[i]={\mathbb {F}}_{9}\) and using the natural order \(\Lambda ={\mathbb {Z}}[i]\oplus {\mathbb {Z}}[i]t\) in \(S_f\), we obtain the nonassociative algebra \(\Lambda /3\Lambda \cong S_{\overline{f}}\) over \({\mathbb {F}}_{3}\) with \(\overline{f}(t)=t^2-(\alpha +\alpha ^3-1)t+\alpha -3\in {\mathbb {F}}_{9}[t;\overline{\sigma }]\) where \(\overline{\sigma }(\alpha )=\alpha ^3\), if \(\alpha \) is a primitive root of \({\mathbb {F}}_{9}\) over \({\mathbb {F}}_{3}\), that is \(\alpha ^2+1=0\). Since
\(\overline{f}\) is reducible in \({\mathbb {F}}_{9}[t;\overline{\sigma }]\). The left ideal generated by \((t+1+\alpha )\) in \(S_{\overline{f}}\) yields a cyclic \((f,\sigma ,0)\)-code of length 2 and dimension one. Taking the pre-image of it under \(\Psi \) it corresponds to a principal left ideal \(\mathcal {I}/3\Lambda \) in \(\Lambda /3\Lambda \).
$$\begin{aligned} \overline{f}(t)=(t-2+\alpha )(t+1+\alpha ), \end{aligned}$$
4.2 Examples involving nonassociative quaternion algebras
Let \(K=\mathbb {Q}(i)\), \(F=\mathbb {Q}\), so that \({\mathcal {O}}_F={\mathbb {Z}}\) and \({\mathcal {O}}_K={\mathbb {Z}}[i]\). The examples given in [19] are special cases of our construction using cyclic algebras. We now consider some algebras which are not associative.
Let \(f(t)=t^2-bt-c\in {\mathbb {Z}}[i][t,\sigma ]\) be irreducible in \(\mathbb {Q}(i)[t;\sigma ]\). This is equivalent to \(\sigma (z)z-bz-c\not =0\) for all \(z\in \mathbb {Q}(i)\) [46, (17)]. In particular, if \(b,c\in {\mathbb {Z}}\) then f(t) is irreducible if \(b^2+4c<0\) (alternatively, if f is an irreducible polynomial in \({\mathbb {R}}\)) by [4, Corollary 2.6]. Suppose that \(\overline{f}(t)=t^2 -\overline{c}\in ({\mathbb {Z}}[i]/{\mathfrak {p}}{\mathbb {Z}}[i])[t;\overline{\sigma }]\) for some maximal ideal \({\mathfrak {p}}\) in \({\mathcal {O}}_F\).
For the natural order \(\Lambda ={\mathbb {Z}}[i]\oplus {\mathbb {Z}}[i]t\), we obtain the (perhaps nonassociative) quaternion algebraIn particular, \(\Lambda /{\mathfrak {p}}\Lambda =({\mathbb {Z}}[i]/{\mathfrak {p}} {\mathbb {Z}}[i])\oplus ({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K)t\) as \(({\mathbb {Z}}[i]/{\mathfrak {p}}{\mathbb {Z}}[i])\)-module.
$$\begin{aligned} \Lambda /{\mathfrak {p}}\Lambda \cong (({\mathbb {Z}}[i]/{\mathfrak {p}}{\mathbb {Z}}[i])/{\mathbb {F}}_{p^j}, \overline{\sigma },\overline{c})=S_{\overline{f}}. \end{aligned}$$
For any choice of \(c\in {\mathbb {Z}}\) such that \(c\not \in {\mathfrak {p}}{\mathbb {Z}}[i]\), \(\overline{f}(t)=t^2-\overline{c}\in {\mathcal {O}}_K[t,\sigma ]\) is reducible.
For \(b=0\) and any choice of \(c\in {\mathbb {Z}}[i]{\setminus }{\mathbb {Z}}\), \(f(t)=t^2-c\in {\mathbb {Z}}[i][t,\sigma ]\) is irreducible in \(\mathbb {Q}(i)[t;\sigma ]\) and thereforea nonassociative quaternion division algebra (for instance, \(ct=(t^2)t\not =t(t^2)=\sigma (c)t\) in A.) We can also write A as the Cayley–Dickson doubling \(\mathrm{Cay}(\mathbb {Q}(i),c)\), defined in the obvious way.
$$\begin{aligned} A=S_f=(\mathbb {Q}(i)/\mathbb {Q},\sigma ,c) \end{aligned}$$
Example 9
Let \(f(t)=t^2-c\in {\mathbb {Z}}[i][t;\sigma ]\), \(c\in {\mathbb {Z}}[i]{\setminus }{\mathbb {Z}}\). Choose any p which remains inert in \(\mathbb {Q}(i)\), then \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K={\mathbb {F}}_{p^{2j}}\), where j is the inertial degree of \({\mathfrak {p}}\) above p, andIf \(\overline{c}\not =0\) this is a division algebra because \(f(t)=t^2-\overline{c}\) is irreducible. Given any principal left ideal \(\mathcal {I}\) of \(\Lambda \) containing p, \(\Psi (\mathcal {I}/{\mathfrak {p}}\Lambda )\) is thus either trivial or all of \(S_{\overline{f}}={\mathbb {F}}_{p^{2j}}[t; \overline{\sigma }]/{\mathbb {F}}_{p^{2j}}[t;\overline{\sigma }] \overline{f}\).
$$\begin{aligned} \Lambda /{\mathfrak {p}}\Lambda \cong ({\mathbb {F}}_{p^{2j}}/{\mathbb {F}}_{p^j},\overline{\sigma }, \overline{c}). \end{aligned}$$
If \(\overline{c}=0\), i.e. when \(c\in {\mathfrak {p}}{\mathcal {O}}_K\) then \(\overline{f}(t)=t^2\) and \(S_{\overline{f}}\) is a commutative associative algebra. There are no \(\overline{\sigma }\)-constacyclic codes since here \(\overline{c}=0\). Thus this algebra cannot be used for lattice encoding of \(\overline{\sigma }\)-constacyclic codes.
E.g., take \(p=3\). If \(c=i\) then \(\Lambda /3\Lambda \cong ({\mathbb {F}}_{9}/{\mathbb {F}}_{3},\overline{\sigma },\overline{i})\) is a nonassociative quaternion division algebra over \({\mathbb {F}}_{3}\) where \(\overline{\sigma }(\alpha )=\alpha ^3\), if \(\alpha \) is a primitive root of \({\mathbb {F}}_{9}\) over \({\mathbb {F}}_{3}\), that is \(\alpha ^2+1=0\). \(\mathcal {I}=(1+i)\Lambda \) satisfies \(3\in \mathcal {I}\cap {\mathcal {O}}_F\) (since \(1-2i\in {\mathbb {Z}}[i]\), so \((1+i)(1-2i)=3\in \mathcal {I}\)). Hence \(\mathcal {I}/3\Lambda \) is a left principal ideal of \(\Lambda /3\Lambda \cong ({\mathbb {F}}_{9}/{\mathbb {F}}_{3},\overline{\sigma },\overline{i})\), generated by \(\Psi ((1+i)+3\Lambda )\), implying \(\mathcal {I}/3\Lambda \cong ({\mathbb {F}}_{9}/{\mathbb {F}}_{3},\overline{\sigma },\overline{i})\). Since here \(\overline{f}\) is irreducible, the only available (and trivial) \(\overline{\sigma }\)-constacyclic code here is the one corresponding to the algebra \(({\mathbb {F}}_{9}/{\mathbb {F}}_{3},\overline{\sigma },\overline{i})\).
If \(c=3i\), then \(\overline{f}(t)=t^2\) and \(\Lambda /3\Lambda \cong ({\mathbb {F}}_{9}/{\mathbb {F}}_{3},\overline{\sigma },0)\) is a commutative associative algebra over \({\mathbb {F}}_{3}\). There are no \(\overline{\sigma }\)-constacyclic codes since \(\overline{c}=0\), hence this example cannot be used for lattice encoding of \(\overline{\sigma }\)-constacyclic codes.
Example 10
Let \(f(t)=t^2-c\in {\mathbb {Z}}[i][t;\sigma ]\), \(c\in {\mathbb {Z}}[i]{\setminus }{\mathbb {Z}}\).
is a nonassociative quaternion algebra over \({\mathbb {F}}_{5}\) withHere, \(\overline{\sigma }(a,b)=(b,a)\) fixes the elements (a, a), \(a\in {\mathbb {F}}_{5}\). The algebra \(\Lambda /5\Lambda \) is a split nonassociative quaternion algebra [57], however for all \(\overline{c}\not =0\), \(\overline{f}\) is irreducible in \(({\mathbb {F}}_{5}\times {\mathbb {F}}_{5})[t;\overline{\sigma }]\), since \(\overline{c}\not \in {\mathbb {F}}_{5} \). In this case, there are no non-trivial divisors of \(\overline{f}\) and hence no non-trivial codes to lift. If \(\overline{ c}=0\) then \(\overline{f}(t)=t^2\) and \(\Lambda /5\Lambda \) a commutative associative algebra. There are no \(\overline{\sigma }\)-constacyclic codes since \(\overline{c}=0\).
and we have the following \({\mathbb {F}}_{2}\)-algebra isomorphism:with \(\overline{f}(t)=t^2-\overline{c}\in ({\mathbb {F}}_2+{\mathbb {F}}_2 v)[t]\), \(c\in {\mathbb {Z}}[i]{\setminus } {\mathbb {Z}}\). For both \(\overline{c}=v\) and \(\overline{c}=v+1\), it is easy to show that \(\overline{f}\) is irreducible, and if \(\overline{c}=0\) again \(\overline{f}=t^2.\) We conclude that \(p=2\) does not yield an algebra which can be employed for lattice encoding.
(i)
Choose any p which splits in \(\mathbb {Q}(i)\), e.g. \(p=5\). Then \((5)=(1+2i)(1-2i)\) means that \({\mathbb {Z}}[i]/5 {\mathbb {Z}}[i]\cong {\mathbb {Z}}[i]/(1-2i) \times {\mathbb {Z}}[i]/(1+2i)\cong {\mathbb {F}}_5\times {\mathbb {F}}_5\) and
$$\begin{aligned} \Lambda /5\Lambda =({\mathbb {F}}_5\times {\mathbb {F}}_5)\oplus ({\mathbb {F}}_5\times {\mathbb {F}}_5) e \end{aligned}$$
$$\begin{aligned} \Lambda /5\Lambda \cong (({\mathbb {F}}_{5}\times {\mathbb {F}}_{5})/{\mathbb {F}}_{5},\overline{\sigma },\overline{c}) =S_{\overline{f}} \text { with } \overline{f}(t)=t^2-\overline{c}\in ({\mathbb {F}}_{5}\times {\mathbb {F}}_{5})[t;\overline{\sigma }]. \end{aligned}$$
(ii)
Choose \(p=2\) which ramifies in \(\mathbb {Q}(i)\). Then \({\mathbb {Z}}[i]/2 {\mathbb {Z}}[i]\cong {\mathbb {F}}_2 +{\mathbb {F}}_2 v=\{0,1,v,v+1\}\) with \(v^2=0\). I.e., \({\mathbb {F}}_2+{\mathbb {F}}_2 v\) is the finite chain ring of characteristic 2, nilpotency index 2 and residue field \({\mathbb {F}}_{2}\). Here \(\overline{\sigma }=id\),
$$\begin{aligned} \Lambda /2\Lambda =({\mathbb {F}}_2+{\mathbb {F}}_2 v) \oplus ({\mathbb {F}}_2+{\mathbb {F}}_2 v) e \end{aligned}$$
$$\begin{aligned} \Lambda /2\Lambda \cong (({\mathbb {F}}_2 +{\mathbb {F}}_2 v)/{\mathbb {F}}_{2},id,\overline{c}) =S_{\overline{f}} \end{aligned}$$
4.3 Nonassociative cyclic algebras of non-prime degree
For a nonassocative cyclic algebra \(A=(K/F,\sigma ,c)\) of prime degree n, A is a division algebra if and only if \(c\in K{\setminus } F\). Examples 9 and 10 demonstrate that this poses a problem when trying to find irreducible \(f(t)=t^n-c\) such that \(\overline{f}(t)=t^n-\overline{ c}\) is reducible and \(0\not =\overline{ c}\), since \(\overline{f}(t)\) is either irreducible, or \(\overline{ c}=0\). This is not the case when n is not prime:
Example 11
Let \(f(t)=t^4-c\). Let \(\omega _{15}\) be a primitive 15th root of unity, \(K=\mathbb {Q}(i,\omega _{15}+\omega _{15}^{-1})\), and \(F=\mathbb {Q}(i)\). Choose \(c\in {\mathcal {O}}_K{\setminus } {\mathcal {O}}_F\) such that \(1,c,c^2,c^3\) are F-linearly independent. Then \(D=(\mathbb {Q}(i,\omega _{15}+\omega _{15}^{-1})/\mathbb {Q}(i),\sigma ,c)\) is a nonassociative cyclic division algebra of degree 4 andis the natural order in D.
$$\begin{aligned} \Lambda ={\mathbb {Z}}[i,\omega _{15}+\omega _{15}^{-1}]\oplus {\mathbb {Z}}[i,\omega _{15}+\omega _{15}^{-1}] t\oplus {\mathbb {Z}}[i,\omega _{15}+\omega _{15}^{-1}]t^2\oplus {\mathbb {Z}}[i,\omega _{15}+\omega _{15}^{-1}]t^3 \end{aligned}$$
Let \({\mathfrak {p}}=( 1+i)\). Then \({\mathfrak {p}}\) is unramified in \(\mathbb {Q}(i,\omega _{15}+\omega _{15}^{-1})\), \({\mathbb {Z}}[i]/( 1+i)\cong {\mathbb {F}}_2\), andis a nonassociative cyclic algebra of degree 4 over \({\mathbb {F}}_2\) which for \(\overline{ c}\not =0\) is never a division algebra, since \(1,c,c^2,c^3\) are always linearly dependent over \({\mathbb {F}}_2\). Hence \(f(t)=t^4-\overline{c}\) is reducible. If \(\overline{c}=1\) then given any principal left ideal \(\mathcal {I}\) of \(\Lambda \) containing \((1+i)\) that is generated by a monic polynomial, \(\Psi (\mathcal {I}/(1+i)\Lambda )\) corresponds to a \(\overline{\sigma }\)-constacyclic code over \({\mathbb {F}}_{16}\).
$$\begin{aligned} \Lambda /{\mathfrak {p}}\Lambda \cong ({\mathbb {F}}_{16}/{\mathbb {F}}_2, \overline{\sigma },\overline{c}) \end{aligned}$$
5 Natural orders in \(S_f\) and their quotients by a prime ideal, II
5.1 The setup
Let K/F be a cyclic Galois extension of number fields of degree n and let \(D=(K/F, \rho , c)\) be a cyclic division algebra over F such that \(c\in {\mathcal {O}}_F^\times \). Let \({\mathcal {D}}=({\mathcal {O}}_K/{\mathcal {O}}_F,\rho , c)\) be the generalized associative cyclic algebra over \({\mathcal {O}}_F\) of degree n such that \({\mathcal {D}} \otimes _{{\mathcal {O}}_F}F=(K/F, \rho , c)=D\). Then \({\mathcal {D}}={\mathcal {O}}_K \oplus {\mathcal {O}}_Ke \oplus \dots \oplus {\mathcal {O}}_K e^{n-1}\) is a natural \({\mathcal {O}}_F\)-order of D, cf. 3.2 or [19].
Let \(\sigma \in \mathrm{Aut}(D)\) and \(\delta \) be a \(\sigma \)-derivation on D, satisfying the following criteria:Suppose \(f(t)=\sum _{i=0}^{m}d_it^i\in {\mathcal {D}}[t;\sigma ,\delta ]\) is monic and irreducible in \(D[t;\sigma ,\delta ]\). Consider the division algebraover \(F_0\). Then the \(S_0\)-algebrais an \(S_0\)-order in \(S_f\) which we call the natural order (this is again usually not maximal). \(\Lambda \) is not uniquely determined even when Rf is not a two-sided ideal. It depends on the choice of the maximal subfield in D which we will assume to be K.
-
\(F_0=F\cap \mathrm{Fix}(\sigma )\cap \mathrm{Const}(\delta )\) is a number field.
-
\(\sigma ({\mathcal {D}})\subset {\mathcal {D}}\) and \(\delta ({\mathcal {D}})\subset {\mathcal {D}}\).
-
\(S_0={\mathcal {O}}_F\cap \mathrm{Fix}(\sigma )\cap \mathrm{Const} (\delta )\) is the ring of integers of \(F_0\) where here \(\sigma \) and \(\delta \) denote the restrictions of \(\sigma \) and \(\delta \) to \({\mathcal {D}}\).
$$\begin{aligned} S_f=D[t;\sigma ,\delta ]/D[t;\sigma ,\delta ] f \end{aligned}$$
$$\begin{aligned} \Lambda ={\mathcal {D}}[t;\sigma ,\delta ]/{\mathcal {D}}[t;\sigma ,\delta ]f \end{aligned}$$
Since f is irreducible in \(D[t;\sigma ,\delta ]\), \(\Lambda \) does not have zero divisors. If \(1,e,\dots ,e^{n-1}\) is the canonical basis of D thenas left \({\mathcal {O}}_K\)-module.
$$\begin{aligned} \Lambda= & {} {\mathcal {O}}_K \oplus {\mathcal {O}}_K e\oplus \dots \oplus {\mathcal {O}}_K e^{n-1} \oplus {\mathcal {O}}_K t\oplus {\mathcal {O}}_K et\\&\oplus \dots \oplus {\mathcal {O}}_K e^{n-1}t \oplus \dots \oplus {\mathcal {O}}_K e^{n-1}t^{m-1} \end{aligned}$$
Let \({\mathfrak {p}}\) be a prime ideal in \(S_0\) such that \({\mathfrak {p}}{\mathcal {O}}_F\) is maximal. Since \(S_0\) lies in the centers of both \({\mathcal {D}}\) and \(\Lambda \), \( {\mathfrak {p}}{\mathcal {D}}\) is a two-sided ideal of \({\mathcal {D}}\) and \({\mathfrak {p}}\Lambda \) is a two-sided ideal of \(\Lambda \). Let \(\pi :{\mathcal {D}}\longrightarrow {\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}}\) be the canonical projection. We have \(\sigma ( {\mathfrak {p}}{\mathcal {D}})\subset {\mathfrak {p}}{\mathcal {D}}\) since \({\mathfrak {p}}\subset \mathrm{Fix}(\sigma )\) and \(\sigma ({\mathcal {D}})\subset {\mathcal {D}}\) by assumption. Thus \(\sigma \) induces a ring homomorphismwith \(\mathrm{Fix}(\overline{\sigma })=\mathrm{Fix}(\sigma )/{\mathfrak {p}}\mathrm{Fix}(\sigma )\) and \(\pi \circ \sigma =\overline{\sigma } \circ \pi \). We also have \(\delta ( {\mathfrak {p}}{\mathcal {D}})\subset {\mathfrak {p}}{\mathcal {D}}\) by assumption, so that \(\delta \) induces a left \(\overline{\sigma }\)-derivation \(\overline{\delta }:{\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}} \longrightarrow {\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}}\) with field of constants \(\mathrm{Const}(\overline{\delta })=\mathrm{Const}(\delta )/{\mathfrak {p}}.\) Letwith \(\overline{F}={\mathcal {O}}_F/{\mathfrak {p}}{\mathcal {O}}_F={\mathbb {F}}_{p^j}\), where j is the inertial degree of \({\mathfrak {p}}\) above p. For any \(g(t)=\sum _{i=0}^{m-1}a_i t^i\in {\mathcal {D}}[t;\sigma ,\delta ]\) define \(\overline{g}(t)=\sum _{i=0}^{m-1}\overline{a_i}t^i\in ({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}})[t;\overline{\sigma }, \overline{\delta }]\) with \(\overline{a_i}= a_i+{\mathfrak {p}}{\mathcal {D}}\). Let \(\overline{f}(t)=\sum _{i=0}^{m}\overline{d_i}t^i\in ({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}})[t;\overline{\sigma }, \overline{\delta }]\) with \(\overline{d_i}= d_i+{\mathfrak {p}}{\mathcal {D}}\), thenSince \(\overline{S_0}=S_0/{\mathfrak {p}} \cong \mathrm{Fix}(\overline{\sigma })\cap \mathrm{Const}(\overline{\delta })\cap \overline{F}\), \(S_{\overline{f}}\) is an algebra over a subfield of \({\mathbb {F}}_{p^j}\).
$$\begin{aligned} \overline{\sigma }:{\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}} \longrightarrow {\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}} \end{aligned}$$
$$\begin{aligned} \overline{S_0}=\mathrm{Fix}(\overline{\sigma })\cap \mathrm{Const}(\overline{\delta })\cap \overline{F} \end{aligned}$$
$$\begin{aligned} S_{\overline{f}}=({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}}) [t;\overline{\sigma },\overline{\delta }]/({\mathcal {D}}/{\mathfrak {p}} {\mathcal {D}}) [t;\overline{\sigma },\overline{\delta }]\overline{f}. \end{aligned}$$
Lemma 12
(i)
The surjective homomorphism of additive groups has kernel \({\mathfrak {p}}\Lambda \).
$$\begin{aligned} \Psi :\Lambda \longrightarrow ({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}})[t;\overline{\sigma }, \overline{\delta }]/ ({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}})[t;\overline{\sigma }, \overline{\delta }]\overline{f},\quad g\mapsto \overline{g} \end{aligned}$$
(ii)
\(\Psi \) induces an \(\overline{S_0}\)-algebra isomorphism
$$\begin{aligned} \Psi :\Lambda /{\mathfrak {p}}\Lambda \cong ({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}})[t;\overline{\sigma }, \overline{\delta }]/({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}}) [t;\overline{\sigma },\overline{\delta }]\overline{f},\quad g +{\mathfrak {p}}\Lambda \mapsto \overline{g} \end{aligned}$$
Proof
(i)
For all \(g=\sum _{i=0}^{m-1}b_it^i\in {\mathfrak {p}}\Lambda \) we have \(\overline{g}=\sum _{i=0}^{m-1}\overline{b_i}t^i=0\) in \(({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}})[t;\overline{\sigma },\overline{\delta }]\), so that \({\mathfrak {p}}\Lambda \subset ker(\Psi )\).
Suppose conversely there is a nonzero \(g\in \Lambda \) such that \(\Psi (g)=\overline{g}=0\), then \(g=hf+r\) in \({\mathcal {D}}[t;\sigma ,\delta ]\) with a nonzero \(r\in {\mathcal {D}}[t;\sigma ,\delta ]\), and so \(\overline{g}=\overline{h}\,\overline{f}+\overline{r}\) with \(\overline{r}=0\) in \(({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}})[t;\overline{\sigma },\overline{\delta }]\). We have \({\mathfrak {p}}\Lambda =\{al\,|\, a\in {\mathfrak {p}},l\in \Lambda \}=\) \(\{\sum _{i=0}^{m-1}a_it^i\,|\, a_i\in {\mathfrak {p}}{\mathcal {D}}\}.\) This implies that \(r\in {\mathfrak {p}}{\mathcal {D}}[t;\sigma ,\delta ]={\mathfrak {p}}\Lambda \).
(ii)
follows from (i). \(\square \)
5.2 A special case
Let \(S_f=(D,\sigma ,d)\) be the \(F_0\)-algebra constructed in Example 2 where now F, L and K be number fields. Suppose that \(c \in {\mathcal {O}}_{F_0}\) and that \(d\in {\mathcal {O}}_L^\times \) or \(d\in {\mathcal {O}}_F^\times \). Then \({\mathcal {D}}=({\mathcal {O}}_K/{\mathcal {O}}_F, \rho , c)\) is an associative cyclic algebra over \({\mathcal {O}}_F\) of degree n such that \({\mathcal {D}} \otimes _{{\mathcal {O}}_F}F=(K/F, \rho , c)=D\) is a division algebra over F. For \(x= x_0 + x_1 e+x_2 e^2 +\dots + x_{n-1}e^{n-1}\in D\), defineSince \(c \in {\mathcal {O}}_{F_0}\), \(\sigma \in \mathrm{Aut}_L(D)\) has order m and restricts to \(\sigma \in \mathrm{Aut}_{{\mathcal {O}}_{L}}({\mathcal {D}})\).
$$\begin{aligned} \sigma (x)=\sigma (x_0) + \sigma (x_1)e + \sigma (x_2)e^2 +\dots + \sigma (x_{n-1})e^{n-1}. \end{aligned}$$
Let \({\mathfrak {p}}\) be a prime ideal in \({\mathcal {O}}_{F_0}\) such that \({\mathfrak {p}}{\mathcal {O}}_F\) is maximal. Thenis an algebra over \(\overline{F_0}={\mathcal {O}}_{F_0}/{\mathfrak {p}}\), with \(\overline{D}={\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}}\) a generalized associative cyclic algebra over \({\mathbb {F}}_{p^j}=\mathrm{Fix}(\overline{\rho })\).
$$\begin{aligned} \Lambda /{\mathfrak {p}}\Lambda \cong ({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}})[t;\overline{\sigma }]/ ({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}}) [t;\overline{\sigma }]\overline{f}\cong (\overline{D}, \overline{\sigma }, \overline{d}) \end{aligned}$$
Example 13
Let \(\omega \) denote the primitive third root of unity, \(\omega _7\) a primitive 7th root of unity and \(\theta = \omega _7 + \omega _7^{-1} = 2 \cos (\frac{2 \pi }{7})\). Put \(F = \mathbb {Q}(\theta )\). Let \(K = F(\omega ) =\mathbb {Q}(\omega , \theta )\) and consider the quaternion division algebra \(D = (K/F, \sigma , -1)\). Note that \(\sigma (\omega ) = \omega ^2\). Let \(L =\mathbb {Q}(\omega )\), so that K / L is a cubic cyclic field extension whose Galois group is generated by the automorphism \(\tau : \omega _7 +\omega _7^{-1} \mapsto \omega _7^2 + \omega _7^{-2}\). Note that \(\omega \in {\mathcal {O}}_L={\mathbb {Z}}[\omega ]\).
The multiplication of the division algebra \(A = (D, \tau , \omega )\) is behind the fully diverse codes employed in [54] (cf. [48]). Here,is a natural order in A.
$$\begin{aligned} \Lambda ={\mathbb {Z}}[\omega _3,\omega _7+\omega _7^{-1}]\oplus {\mathbb {Z}}[\omega _3,\omega _7+\omega _7^{-1}] e\oplus {\mathbb {Z}}[ \omega _3,\omega _7+\omega _7^{-1}]e^2\oplus \dots \end{aligned}$$
Let \(p=2\), then \({\mathfrak {p}}\) is a prime ideal in \({\mathcal {O}}_F={\mathbb {Z}}[\omega _3]\) which remains prime in \({\mathcal {O}}_K={\mathbb {Z}}[\omega _3,\omega _7+\omega _7^{-1}]\) and \({\mathbb {Z}}[i]/{\mathfrak {p}}\cong {\mathbb {F}}_4\). \({\mathfrak {p}}\) is inert in \(K=\mathbb {Q}(\omega _3,\omega _7+\omega _7^{-1})\). Nowis a split quaternion algebra over \({\mathbb {F}}_8\). Thuswhere \(\overline{\omega }\in {\mathbb {Z}}[\omega ]/2{\mathbb {Z}}[\omega ]\).
$$\begin{aligned} {\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}}=({\mathbb {F}}_{64}/{\mathbb {F}}_8,\overline{\sigma },-1)\cong \mathrm{Mat}_2({\mathbb {F}}_8) \end{aligned}$$
$$\begin{aligned} \Lambda /{\mathfrak {p}}\Lambda \cong (\mathrm{Mat}_2({\mathbb {F}}_8),\overline{\tau },\overline{\omega }) \end{aligned}$$
6 Lattice encoding of cyclic \((f,\sigma ,\delta )\)-codes over \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\), II
We continue to assume the setup from Sect. 5.1.
6.1 A second generalization of Construction A
Let \(\mathcal {I}\) be a principal left ideal of \(\Lambda \) generated by a monic polynomial g(t), such that \({\mathfrak {p}}\subset \mathcal {I}\cap S_0\). Then \(\mathcal {I}/{\mathfrak {p}}\Lambda \) is a non-zero principal left ideal of \(\Lambda /{\mathfrak {p}}\Lambda \) and \(\Psi (\mathcal {I}/{\mathfrak {p}}\Lambda )\) is a principal left ideal of \(S_{\overline{f}}\) generated by the monic polynomial \(\Psi (g+{\mathfrak {p}}\Lambda )=\overline{g}\). That means \(\Psi (\mathcal {I}/{\mathfrak {p}}\Lambda )\) corresponds to an \((\overline{f},\overline{\sigma },\overline{\delta })\)-code \({\mathcal {C}}\) over \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\).
In particular, if we choose \(\delta =0\) and f(t) such that \(\overline{f}(t)=t^m-\overline{ c}\in {\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}}\) with \(\overline{c}\) non-zero, then \(\Psi (\mathcal {I}/{\mathfrak {p}}\Lambda )\) is a \(\overline{\sigma }\)-constacyclic code over \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\).
If \(\overline{f}\) is irreducible and \({\mathcal {D}}/{\mathfrak {p}}{\mathcal {D}}\) is a division algebra, then the algebra \(S_{\overline{f}}\) is simple. Then any non-zero code \({\mathcal {C}}\) must have length m and dimension m (and correspond to the whole algebra), whereas whenever \(\overline{f}\) is reducible, \({\mathcal {C}}\) respectively \(\Psi (\mathcal {I}/{\mathfrak {p}}\Lambda )\) corresponds to a right divisor \(\overline{g}\) of \(\overline{f}\) and has dimension \(m-\mathrm{deg}(\overline{g})\). Letbe the canonical projection \(\Lambda \longrightarrow \Lambda /{\mathfrak {p}}\Lambda \) composed with \(\Psi \). We know that \({\mathcal {O}}_K\) is a free \({\mathbb {Z}}\)-module of rank \(n[F:\mathbb {Q}]\). Thereforeis a \({\mathbb {Z}}\)-module of dimension \(N=n^2m[F:\mathbb {Q}]\). The embedding of this lattice into \({\mathbb {R}}^N\) is canonically determined by considering \(S_f\otimes _{\mathbb {Q}}{\mathbb {R}}\). Again all works exactly as explained in [19, Section 3.3] (since associativity is not relevant for the argument). The construction of L can again be seen as a second (nonassociative) variation of the non-commutative Construction A in [18].
$$\begin{aligned} \rho : \Lambda \longrightarrow \Lambda /{\mathfrak {p}}\Lambda \longrightarrow \Psi (\Lambda /{\mathfrak {p}}\Lambda ) \end{aligned}$$
$$\begin{aligned} L=\rho ^{-1}({\mathcal {C}})=\mathcal {I} \end{aligned}$$
In this way we can construct a lattice L in \({\mathbb {R}}^N\) from the linear \((\overline{f},\overline{\sigma },\overline{\delta })\)-code \({\mathcal {C}}\) over a finite ring.
Note that even if \(R=D[t;\sigma ,\delta ]\) is isomorphic to \(D[t;\sigma ]\) or \(D[t;\delta ]\), like when \(\sigma \) or \(\delta \) are inner, we conjecture that the codes/lattices we obtain from using different ways to write R can be substantially different in performance, similarly as the examples obtained in [8], where some of the codes obtained by working with the general skew polynomial ring \({\mathbb {F}}_q[t;\sigma ,\delta ]\) have a better distance bound than the ones obtained with \(\delta =0\).
6.2 Space–time block codes
We now apply the above considerations to space–time block coding (cf. 2.4).
Example 14
Let \(A=(K/F,\sigma ,c)\), \(c\in {\mathcal {O}}_K\) non-zero, be a nonassociative cyclic division algebra over F of degree m with \(c\in {\mathcal {O}}_K\). Take the natural order \(\Lambda ={\mathcal {O}}_K[t;\sigma ]/{\mathcal {O}}_K[t;\sigma ]f\), and let \(a = a_0 + a_1t + \cdots + a_{m-1}t^{m-1},\,b = b_0 + b_1t + \cdots + b_{m-1}t^{m-1}\in \Lambda \). If we identify a with the vector \((a_0, a_1, \ldots , a_{m-1})\), we can express \(R_a\in \mathrm{End}_{{\mathcal {O}}_K}(\Lambda )\) as an \(m \times m\)-matrix M(a) with entries in \({\mathcal {O}}_K\):If we identify b with \((b_0, b_1, \ldots , b_{m-1})\), the right multiplication with a in \(\Lambda \) is given by the matrix multiplication \(b\cdot a=bM(a) \). The family of matrices \(\{M(a)\,|\, 0\not =a\in A\}\) is a fully diverse linear space–time block code \({\mathcal {C}}\).
$$\begin{aligned} M(a)= \left[ \begin{array}{ccccc} a_0 &{} c \sigma (a_{m-1})&{} c \sigma ^2(a_{m-2}) &{} \cdots &{} c \sigma ^{m-1}(a_1) \\ a_1 &{} \sigma (a_0) &{} c\sigma ^2(a_{m-1}) &{} \cdots &{} c \sigma ^{m-1}(a_{2}) \\ a_2 &{} \sigma (a_1) &{} \sigma ^2(a_0) &{} \cdots &{} c \sigma ^{m-1}(a_3)\\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{m-1} &{} \sigma (a_{m-2}) &{} \sigma ^2(a_{m-3}) &{} \cdots &{} \sigma ^{m-1}(a_0) \end{array} \right] ^T . \end{aligned}$$
(1)
Let \({\mathfrak {p}}\subset {\mathcal {O}}_{F}\) be a maximal ideal. Then \(\rho : \Lambda \longrightarrow \Lambda /{\mathfrak {p}}\Lambda \longrightarrow \Psi (\Lambda /{\mathfrak {p}}\Lambda )= (({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K)/{\mathbb {F}}_{p^f}, \overline{\sigma },\overline{c}).\) Hence \(L=\rho ^{-1}({\mathcal {C}})=\mathcal {I}\) is a fully diverse space–time block code over \({\mathcal {O}}_K\) which is a \({\mathbb {Z}}\)-lattice whose embedding into \({\mathbb {R}}^{m}\) is canonically determined by \(A\otimes _{\mathbb {Q}}{\mathbb {R}}\).
Nonassociative cyclic division algebras as above can be employed to obtain fully diverse multiple-input double-output codes [55]. The algebras \(A=(D,\sigma ^{-1},d)\) we consider next are used for the systematic space–time block code constructions of the fast-decodable iterated codes in [37, 49, 50, 54].
Example 15
Let \(A=(D,\sigma ,d)\) be a division algebra of degree n and \(d\in {\mathcal {O}}_L\) or \(d\in {\mathcal {O}}_F\).
For \(x=x_0+x_1t+x_2t^2+\cdots +x_{m-1}t^{m-1}\), \(y=y_0+y_1t+y_2t^2+\cdots +y_{m-1}t^{m-1}\in \Lambda \), \(x_{i}, y_i \in {\mathcal {D}}\), represent x as \((x_0, x_1, \ldots , x_{m-1})\) and y as \((y_0, y_1, \ldots , y_{m-1})\). Then \(R_x\in \mathrm{End}_{\mathcal {D}}(\Lambda )\) is given by the \(m \times m\)-matrixwith entries in \({\mathcal {D}}\). We can write the multiplication in \(\Lambda \) as \(y\cdot x= yM(x)\). Now substitute the right regular representation \(\gamma (d)\) in \({\mathcal {D}}\) for d in M(x) and the right regular representation \(\gamma (x_i)\) in \({\mathcal {D}}\) for each entry \(x_i\) in M(x). This way we obtain a block matrixwhere \(\sigma (\gamma (x_i))\) means we apply \(\sigma \) to each entry of the \(n\times \)-matrix \(\gamma (x_i)\). Products are the usual matrix products. This is an \(mn \times mn\) matrix with entries in \({\mathcal {O}}_K\). It represents right multiplication in \(\Lambda \). Writing elements in \(\Lambda ={\mathcal {O}}_K \oplus {\mathcal {O}}_K e\oplus \dots \oplus {\mathcal {O}}_K e^{n-1}t^{m-1}\) as row vectors of length mn with entries in \({\mathcal {O}}_K\), we obtain \(y\cdot x = y\, \gamma (M(x))\).
$$\begin{aligned} M(x) = \left[ \begin{array}{ccccc} x_0 &{} d \sigma (x_{m-1})&{} d \sigma ^{2}(x_{m-2}) &{} \cdots &{} d \sigma ^{m-1}(x_1) \\ x_1 &{} \sigma (x_0) &{} d \sigma ^{2}(x_{m-1}) &{} \cdots &{} d \sigma ^{m-1}(x_{2}) \\ x_2 &{} \sigma (x_1) &{} \sigma ^{2}(x_0) &{} \cdots &{} d \sigma ^{m-1}(x_3)\\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ x_{m-1} &{} \sigma (x_{m-2}) &{} \sigma ^{2}(x_{m-3}) &{} \cdots &{} \sigma ^{m-1}(x_0) \end{array} \right] ^T \end{aligned}$$
$$\begin{aligned} \gamma (M(x)) := \left[ \begin{array}{cccc} \gamma (x_0) &{} \gamma (d) \sigma (\gamma (x_{m-1}))&{} \cdots &{} \gamma (d) \sigma ^{m-1}(\gamma (x_1)) \\ \gamma (x_1) &{} \sigma (\gamma (x_0)) &{} \cdots &{} \gamma (d) \sigma ^{m-1}(\gamma (x_{2})) \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \gamma (x_{m-1}) &{} \sigma (\gamma (x_{m-2})) &{} \cdots &{} \sigma ^{m-1} (\gamma (x_0)) \end{array} \right] ^T \end{aligned}$$
The family of matrices \(\{ \gamma (M(x))\}\) is a fully diverse linear space–time block code \({\mathcal {C}}\). In particular, if \(d\in {\mathcal {O}}_F\), then \(\det (\gamma (M(x))) \in {\mathcal {O}}_F\) ([37], [49, Remark 5]) and if \(d\in {\mathcal {O}}_L\), thenand \(\det (\gamma (M(x))) \in L\cap {\mathcal {O}}_K={\mathcal {O}}_L\) ([54], [49, Lemma 19]).
$$\begin{aligned} \gamma (M(x)) = \begin{bmatrix} \gamma (x_0)&d \sigma (\gamma (x_{n-1}))&d \sigma ^{2}(\gamma (x_{n-2}))&\cdots&d \sigma ^{m-1}(\gamma (x_1)) \\ \gamma (x_1)&\sigma (\gamma (x_0))&d \sigma ^{2}(\gamma (x_{n-1}))&\cdots&d \sigma ^{m-1}(\gamma (x_{2})) \\ \gamma (x_2)&\sigma (\gamma (x_1))&\sigma ^{2}(\gamma (x_0))&\cdots&d \sigma ^{m-1}(\gamma (x_3))\\ \vdots&\vdots&\vdots&\ddots&\vdots \\ \gamma (x_{n-1})&\sigma (\gamma (x_{n-2}))&\sigma ^{2}(\gamma (x_{n-3}))&\cdots&\sigma ^{m-1}(\gamma (x_0)) \end{bmatrix}^T \end{aligned}$$
Let \({\mathfrak {p}}\subset {\mathcal {O}}_{F_0}\) be a prime ideal such that \({\mathfrak {p}}{\mathcal {O}}_F\) is maximal. ThenHere, \(L=\rho ^{-1}({\mathcal {C}})=\mathcal {I}\) induces a fully diverse STBC over \({\mathcal {O}}_K\) which is a \({\mathbb {Z}}\)-lattice whose embedding into \({\mathbb {R}}^N\), \(N=mn^2\), is canonically determined by \(A\otimes _{\mathbb {Q}}{\mathbb {R}}\).
$$\begin{aligned} \rho : \Lambda \longrightarrow \Lambda /{\mathfrak {p}}\Lambda \longrightarrow \Psi (\Lambda /{\mathfrak {p}}\Lambda ) =(\overline{D}, \overline{\sigma }, \overline{d}). \end{aligned}$$
Remark 16
The explanations in [19, Section 5.2, 5.3] hold analogously for our generalization of Construction A in Sect. 4.1 and the examples here, and show the potential of the construction for coset coding used in space–time block coding, in particular for wiretap space–time block coding, but also for linear codes over finite rings. Moreover, the matrix generating a cyclic \((f,\sigma ,\delta )\)-code \({\mathcal {C}}\subset S^m\) represents the right multiplication \(R_g\) in \(S_f\) and is a control matrix of \({\mathcal {C}}\) [51].
7 Conclusion
We presented a method how to construct a lattice from a suitable \((f,\sigma ,\delta )\)-code defined over a finite ring which can be seen as a generalization of the classical Construction A. This can be summarized as follows: Let D be a cyclic division algebra over F which is already defined over \({\mathcal {O}}_F\), or a Galois field extension and f defined over its ring of integers. Take the additional assumptions on \(\sigma \) and \(\delta \) as given in the corresponding previous sections.If we want to apply this construction to space time block coding instead, we substitute the last step with:If desired, this method can be extended to work for any Noetherian integral domain and central simple algebra over its quotient field. It can be applied for coset coding and wiretap coding analogously as described in [19, Sections 5.2, 5.3].
-
Choose some monic skew polynomial \(f\in {\mathcal {D}}[t;\sigma ,\delta ]\) (resp., \(f\in {\mathcal {O}}_K[t;\sigma ,\delta ]\) in the field case) which is irreducible in \(D[t;\sigma ,\delta ]\).
-
Take a natural order \(\Lambda \) of \(S_f\).
-
Choose a prime ideal \({\mathfrak {p}}\) in \(S_0\). This yields the finite ring \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\) you consider the code \({\mathcal {C}}\) to be defined over. \(\overline{f}\) must be reducible in \(({\mathcal {D}}/{\mathfrak {p}} {\mathcal {D}})[t;\overline{\sigma },\overline{\delta }]\).
-
Choose a principal left ideal \(\mathcal {I}\) of \(\Lambda \) generated by a monic polynomial, such that \({\mathfrak {p}}\subset \mathcal {I}\cap S_0\).
-
\(\Psi (\mathcal {I}/{\mathfrak {p}}\Lambda )\) corresponds to an \((\overline{f},\overline{\sigma },\overline{\delta })\)-code \({\mathcal {C}}\) over \({\mathcal {O}}_K/{\mathfrak {p}}{\mathcal {O}}_K\), and \(L=\rho ^{-1}({\mathcal {C}})=\mathcal {I}\) is a \({\mathbb {Z}}\)-lattice whose embedding into \({\mathbb {R}}^N\) is canonically determined by \(S_f\otimes _{\mathbb {Q}}{\mathbb {R}}\).
-
Take the matrix representing right multiplication in \(\Lambda \) and let \({\mathcal {C}}\) be the associated space–time block code. Then \(\rho ^{-1}({\mathcal {C}}) =\mathcal {I}\) is a fully diverse space–time block code which is a \({\mathbb {Z}}\)-lattice.
It would be interesting to investigate which properties of \({\mathcal {C}}\) carry over to the lattice STBC L and find examples of well performing coset codes.
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