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.
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.
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.
Throughout the paper we will put a special emphasis on the nonassociative cyclic algebras
\((K/F,\sigma ,c)\) employed in [
55], and on the generalized nonassociative cyclic algebras
\((D, \sigma , d)\), since these are used for iterated space–time block codes [
48,
49].
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 if
-
f has an invertible leading coefficient,
-
g has an invertible leading coefficient,
-
S is a domain.
An 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
\(f=gh\).
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 map
$$\begin{aligned} \gamma : S_f \rightarrow \mathrm{End}_K(S_f), a\mapsto R_a \end{aligned}$$
induces an injective
S-linear map
$$\begin{aligned} \gamma : S_f \rightarrow \mathrm{Mat}_m(S), a\mapsto R_a \mapsto M(a). \end{aligned}$$
This fact is exploited when designing space–time block codes which employ one of the following two special cases of algebras:
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
$$\begin{aligned} (a_0,\dots ,a_{m-1})\in {\mathcal {C}}\Rightarrow (\sigma (a_{m-1})c,\sigma (a_0),\dots ,\sigma (a_{m-2}))\in {\mathcal {C}}. \end{aligned}$$
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.
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
Let
$$\begin{aligned} \rho : \Lambda \longrightarrow \Lambda /{\mathfrak {p}}\Lambda \longrightarrow \Psi (\Lambda /{\mathfrak {p}}\Lambda ) \end{aligned}$$
be 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
\(n[F:\mathbb {Q}]\). Then
$$\begin{aligned} L=\rho ^{-1}({\mathcal {C}})=\mathcal {I} \end{aligned}$$
is 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].
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.
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 algebra
$$\begin{aligned} \Lambda /{\mathfrak {p}}\Lambda \cong (({\mathbb {Z}}[i]/{\mathfrak {p}}{\mathbb {Z}}[i])/{\mathbb {F}}_{p^j}, \overline{\sigma },\overline{c})=S_{\overline{f}}. \end{aligned}$$
In 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.
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 therefore
$$\begin{aligned} A=S_f=(\mathbb {Q}(i)/\mathbb {Q},\sigma ,c) \end{aligned}$$
a 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.
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:
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.
-
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 \(S_f\otimes _{\mathbb {Q}}{\mathbb {R}}\).
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].
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.