7.1 Primitive idempotent tables \( {\rm M}^{n,q, - 1} \)
The zeros of the irreducible polynomial
\( P_{t}^{ - 1} (x) \) are written in the
\( \alpha ,\zeta \)-representation, i.e. as
\( \alpha \zeta^{i} \), with
\( \zeta = \alpha^{2} \) and
\( i \in C_{t}^{n,q, - 1} \). Since for binary codes there is no distinction between cyclic and negacyclic codes, we assume that
q is odd. At the end of Sect.
4 we defined
\( C_{{t^{*} }}^{n,q, - 1} = C_{n - t - 1}^{n,q, - 1} \) as the conjugated negacyclotomic coset of
\( C_{t}^{n,q, - 1} \),
\( t \in T^{n,q, - 1} \). The integer
\( t* = n - t - 1 \) is an integer in
\( [0,n - 1] \), and we assume that it is an element of
\( T^{n,q, - 1} \). As a consequence of Theorem 12 (i) we have
\( m_{{t^{*} }}^{ - 1} = m_{t}^{ - 1} \). Similarly as in the case
\( \lambda = 1 \), we define
\( P_{{t^{*} }}^{ - 1} (x) \) as the conjugated irreducible polynomial of
\( P_{t}^{ - 1} (x) \) and
\( \theta_{{t^{*} }} (x) \) as the conjugated primitive idempotent of
\( \theta_{t} (x) \). Just as in the case
\( \lambda = 1 \) in Sect.
6,
\( P_{{t^{*} }}^{ - 1} (x) \) is the monic reciprocal of
\( P_{t}^{ - 1} (x) \) (cf. Theorem 2 (vi)). We now present a number of properties of the matrix elements
\( \mu_{s,t} \) of
\( {\rm M}^{n,q, - 1} \). According to Theorem 19 (i) these are equal to the sums of the
\( s \)-powers of the zeros of
\( P_{t}^{ - 1} (x) \). Similar properties for the matrix elements of
\( \varXi^{n,q, - 1} \) can be obtained by
\( \xi_{s}^{t} = ( - 1)^{{a_{s} }} \mu_{s,t} /n \).
In [
25] it was shown, by a few examples, that in the case of cyclic codes the idempotent table or parts of it can sometimes be determined without explicit knowledge of the irreducible polynomials contained in
\( x^{n} - 1 \). Here, we shall show that the same is possible sometimes for negacyclic codes.
7.2 Blocks of conjugated negacyclonomials and idempotents
In order to define r-conjugated negacyclonomials, negacyclotomic cosets and idempotent generators, we first prove the following theorem.
We call
\( C_{rt + a}^{n,q, - 1} \) the
r-
conjugate of
\( C_{t}^{n,q, - 1} \) for any
\( r \in U_{n} \). Consistently, the irreducible polynomial
\( P_{rt + a}^{ - 1} (x) \) is called the
r-
conjugate of
\( P_{t}^{ - 1} (x) \) and the corresponding primitive idempotent
\( \theta_{rt + a} (x) \) the
r-
conjugate of
\( \theta_{t} (x) \). For
\( r = - 1 \) one gets the normal conjugated objects as defined earlier in this section. The definitions of
r-
self conjugateness are completely similar to those in the case
\( \lambda = 1 \). Analogously to Eq. (
16), there exits a simple relationship between
\( \theta_{t} (x) \) and its
r-conjugate.
Next, we introduce the notation (cf. (
17) and (
18))
$$ Cy^{n,q, - 1} : = \{ c_{s}^{ - 1} (x) |s \in S^{n,q, - 1} \} $$
(23)
for the set of all negacyclonomials and
$$ Id^{n,q, - 1} : = \{ \theta_{t} (x) |t \in T^{n,q, - 1} \} . $$
(24)
for the set of all primitive idempotent generators. Just like in Sect.
6, the group
\( U_{n} \) induces a permutation group
\( G^{\prime} \) on the set (
23), while the subgroup
\( H: = \langle q \rangle \) of
\( U_{n} \) contains all elements which induce the identity permutation. Because of the one-one correspondence between negacyclotomic cosets and irreducible polynomials, it follows from Theorem 29 (ii) that
\( U_{n} \) also induces a permutation group
\( G^{\prime\prime} \) on the set
\( Id^{n,q, - 1} \) of (
24).
We next define, as the negacyclic counterpart of (
20), for each positive divisor of
d ≤
n of
n$$ T_{d}^{n,q, - 1} : = \{ t \in T^{n,q, - 1}| (n,2t + 1) = d\} . $$
(25)
Underlying this definition, is that
\( (n,2t + 1) = d \) implies
\( \left( {n,2t^{\prime} + 1} \right) = d \) for
\( t^{\prime}: = tq + (q - 1)/2 \), making the particular choice of
t, as index of some negacyclotomic coset, irrelevant (cf. Eq. (
6) with
\( l = (q - 1)/2 \). It also makes clear that the union of negacyclotomic cosets whose indices are in (
25) contains all integers
\( i \in [0,n - 1] \) with
\( (n,2i + 1) = d \), and also that
$$ T^{n,q, - 1} = \bigcup\limits_{d} {T_{d}^{n,q, - 1} } . $$
(26)
Just as in Sect.
6, the orbits of
\( G^{\prime\prime} \) in the set (
24) are called
blocks of idempotents. In Sect.
6 we defined the transformations
\( C_{s}^{n,q} \to C_{rs}^{n,q} \) and correspondingly
\( c_{s} (x) \to c_{rs} (x) \) for
\( r \in U_{n} \). In the next we shall restrict the first transformation to those cyclotomic cosets which correspond to a negacyclonomial
\( c_{s}^{ - 1} (x) \). For odd
n this concerns all
\( C_{s}^{n,q} \) (cf. Theorem 13 (i) with
\( k = 2 \)), but for even
n this is not always true. Suppose
\( c_{s}^{ - 1} (x) \) is a negacyclonomial and let
n be even. Then
\( r \in U_{n} \) is odd, and by applying Theorem 13 (i) again, it appears that
\( c_{rs}^{ - 1} (x) \) is also a negacyclonomial. So, for all
n the transformation
\( s \to rs \) defines a permutation on the set (
23). The orbits are called
blocks of negacyclonomials. We are ready now to formulate and to prove the analogue of Theorem 25 for primitive idempotents of negacyclic codes.
There are six negacyclotomic cosets, \( C_{0}^{20,3, - 1} = (0,1,4,13) \), \( C_{2}^{20,3, - 1} = (2,7) \), \( C_{3}^{20,3, - 1} = (3,10,11,14) \), \( C_{5}^{20,3, - 1} = (5,16,9,8) \), \( C_{6}^{20,3, - 1} = (6,19,18,15) \) and \( C_{12}^{20,3, - 1} = (12,17) \).
Now,
\( P_{0}^{ - 1} (x) = x^{4} + x^{2} + x + 1 \) is an irreducible divisor of
\( x^{20} + 1 \). Let
\( \alpha \) be a zero of this polynomial of order 40. The other five irreducible factors are
\( P_{2}^{ - 1} (x) = x^{2} - x - 1 \),
\( P_{3}^{ - 1} (x) = x^{4} + x^{2} - x + 1 \),
\( P_{5}^{ - 1} (x) = x^{4} - x^{3} + x^{2} + 1 \),
\( P_{6}^{ - 1} (x) = x^{4} + x^{3} + x^{2} + 1 \) and
\( P_{12}^{ - 1} (x) = x^{2} - x - 1 \), which have respectively
\( \alpha^{5} \),
\( \alpha^{7} \),
\( \alpha^{11} \),
\( \alpha^{13} \) and
\( \alpha^{25} \) as zeros. Furthermore, the six negacyclonomials are
\( c_{0}^{ - 1} (x) = 1 \),
\( c_{1}^{ - 1} (x) = x^{1} + x^{3} + x^{9} - x^{7} \),
\( c_{5}^{ - 1} (x) = x^{5} + x^{15} \),
\( c_{2}^{ - 1} (x) = x^{2} + x^{6} + x^{18} + x^{14} \),
\( c_{4}^{ - 1} (x) = x^{4} + x^{12} - x^{16} - x^{8} \) and
\( c_{11}^{ - 1} (x) = x^{11} - x^{13} + x^{19} + x^{17} \). It follows that we can define the index sets
\( S^{20,3, - 1} = \{ 0,1,2,4,5,11\} \) and
\( T^{20,3, - 1} = \{ 0,2,3,5,6,12\} \). It will be obvious that
\( c_{1}^{ - 1} (x) \) and
\( c_{11}^{ - 1} (x) \) are each other’s conjugate, while all other negacyclonomials are self conjugated. In order to determine the elements of the table
\( {\rm M}^{20,3, - 1} \), we also need the irreducible polynomials contained in
\( x^{20} - 1 \). These are
\( P_{0} (x) = x - 1 \),
\( P_{1} (x) = x^{4} - x^{2} + x + 1 \),
\( P_{10} (x) = x + 1 \),
\( P_{5} (x) = x^{2} + 1 \),
\( P_{4} (x) = x^{4} + x^{3} + x^{2} + x + 1 \),
\( P_{2} (x) = x^{4} - x^{3} + x^{2} - x + 1 \) and
\( P_{11} (x) = x^{4} + x^{3} - x + 1 \). The indices refer to the
\( \zeta \)-representation of the zeros, based on the zero
\( \zeta ( = \alpha^{2} ) \) of
\( P_{1} (x) \), which has order
\( 20 \). By applying the rule that
\( \mu_{s,t} \) is equal to the sum of the
\( s \)-powers of the zeros of
\( P_{t}^{ - 1} (x) \) we find for the idempotent table
\( {\rm M}^{20,3, - 1} \), the rows of which are indexed respectively by 0, 1, 2, 4, 11, 5 and the columns by 0, 2, 3, 6, 12, 5.
$$ {\rm M}^{20,3, - 1} = \left[ {\begin{array}{*{20}c} 1 & 2 & 1 & 1 & 2 & 1 \\ 0 & 1 & 0 & 2 & 2 & 1 \\ 1 & 0 & 1 & 2 & 0 & 2 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 2 & 0 & 2 & 0 \\ 2 & 2 & 1 & 1 & 1 & 2 \\ \end{array} } \right]. $$
Next, by using relation (
14), we derive the related table
$$ \varXi^{20,3, - 1} = \left[ {\begin{array}{*{20}c} 2 & 1 & 2 & 2 & 1 & 2 \\ 1 & 1 & 2 & 0 & 2 & 0 \\ 1 & 0 & 1 & 2 & 0 & 2 \\ 2 & 2 & 2 & 2 & 2 & 2 \\ 0 & 1 & 0 & 2 & 2 & 1 \\ 2 & 2 & 1 & 1 & 1 & 2 \\ \end{array} } \right] . $$
We collect the weights \( w_{s} \), \( s \in S^{20,3, - 1} \), in a weight vector \( \sigma = (1,2,2,1,2,1) \in GF(3)^{6} \), and similarly the weights \( 1/m_{t}^{ - 1} \), \( t \in T^{20,3, - 1} \), in a weight vector \( \tau = (1,2,1,1,2,1) \in GF(3)^{6} \). With the help of these vectors one easily can verify the orthogonality relations in this case. The columns of \( \varXi^{20,3, - 1} \) provide us with the coefficients \( \xi_{s}^{t} \) in the expressions \( \theta_{t} (x) = \sum\nolimits_{{s \in S^{20,3, - 1} }} {\xi_{s}^{t} c_{s}^{ - 1} (x)} \) for the primitive idempotents. These results are confirmed by the general method of Theorem 3, with \( h(x): = P_{t}^{ - 1} (x) \) and \( g(x) = (x^{20} + 1)/h(x) \). There are five non-empty index subsets \( S_{1}^{20,3, - 1} = \{ 1,11\} \), \( S_{2}^{20,3, - 1} = \{ 2\} \), \( S_{4}^{20,3, - 1} = \{ 4\} \), \( S_{5}^{20,3, - 1} = \{ 3\} \) and \( S_{20}^{20,3, - 1} = \{ 0\} \), and so there are that many blocks of negacyclonomials. On the other hand, there are two non-empty index subsets \( T_{1}^{20,3, - 1} = \{ 0,3,5,6\} \) and \( T_{5}^{20,3, - 1} = \{ 2,12\} \). Hence, there are only two blocks of idempotents, i.e. \( B_{1}^{Id} \) containing \( 2\varphi (20/1)/m_{0}^{ - 1} = 16/4 = 4 \) elements, and \( B_{5}^{Id} \) with \( 2\varphi (20/5)/m_{2}^{ - 1} = 4/2 = 2 \) elements. Finally, we remark that there are four self conjugated negacyclonomials, whereas there are no self conjugated negacyclotomic cosets and therefore no self conjugated idempotent generators. This observation shows that Theorem 25 (iv) is not always true in the negacyclic case. Finally, we illustrate Theorem 30 by two small examples. For \( r = 11 \) the transformation \( t \to rt + (r - 1)/2 \) yields the following permutation of primitive idempotents \( \theta_{0} (x) \to \theta_{5} (x) \), \( \theta_{2} (x) \to \theta_{2} (x) \), \( \theta_{3} (x) \to \theta_{6} (x) \), \( \theta_{6} (x) \to \theta_{3} (x) \), \( \theta_{12} (x) \to \theta_{12} (x) \), \( \theta_{5} (x) \to \theta_{0} (x) \). One can easily verify that the transformation \( \theta_{t} (x) \to \theta_{t} (x^{11} ) \) gives the same permutation. One can accomplish this by applying the relations \( c_{s}^{ - 1} (x^{11} ) = c_{s}^{ - 1} (x) \), \( s \in \{ 0,4,5\} \), \( c_{2}^{ - 1} (x^{11} ) = - c_{2}^{ - 1} (x) \), \( c_{1}^{ - 1} (x^{11} ) = c_{11}^{ - 1} (x) \)and \( c_{11}^{ - 1} (x^{11} ) = c_{1}^{ - 1} (x) \) to the expression for \( \theta_{t} (x) \) as follows from the table \( \varXi^{20,3, - 1} \). In a similar way one can verify that \( \theta_{3t + 1} (x) = \theta_{t} ( - x^{7} ) \) for all \( t \in T^{20,3, - 1} \), by using \( c_{s}^{ - 1} ( - x^{3} ) = c_{s}^{ - 1} (x) \), \( s \in \{ 0,2,4\} \), and \( c_{s}^{ - 1} ( - x^{3} ) = - c_{s}^{ - 1} (x) \), \( s \in \{ 1,5,11\} \). Since \( 11.11 = 1 \) mod \( 40 \) and 7.3 = 1 mod 20, these results are in agreement with Theorem 30. □
For more examples of primitive idempotents of constacyclic and negacyclic codes we refer to [
10,
16,
22,
26‐
28].