On the duality and the direction of polycyclic codes

Abstract

Polycyclic codes are ideals in quotients of polynomial rings by a principal ideal. Special cases are cyclic and constacyclic codes. A MacWilliams relation between such a code and its annihilator ideal is derived. An infinite family of binary self-dual codes that are also formally self-dual in the classical sense is exhibited. We show that right polycyclic codes are left polycyclic codes with different (explicit) associate vectors and characterize the case when a code is both left and right polycyclic for the same associate polynomial. A similar study is led for sequential codes.

On polycyclic codes over a finite chain ring

“…An S-linear code of length n is right polycyclic [1,9] with associate vector a (or simply, right a-cyclic) if it is invariant by right multiplication by the matrix D a . Also in [1,9] they define a code as left b-cyclic if it is invariant by right multiplication by the matrix E b . It is straight forward (adapting the proof in [1,Theorem 4.…”

“…Also in [1,9] they define a code as left b-cyclic if it is invariant by right multiplication by the matrix E b . It is straight forward (adapting the proof in [1,Theorem 4. ] to finite chain rings) to see that any right a-cyclic code where the determinant…”

“…Still polycyclic codes never enjoyed the same popularity that constacyclic codes have because the dual of some polycyclic code is not polycyclic. In [1], an alternate duality is introduced in order to adress this issue.…”

Long quasi-polycyclic <inline-formula><tex-math id="M1">\begin{document}$t-$\end{document}</tex-math></inline-formula> CIS codes

“…For a general polynomial h(x) ∈ F q [x], we will call the codes in R t quasipolycyclic (QPC). This kind of codes have been recently studied in [4] (see also [2]). If h(x) = x n − 1, then the related codes are quasi-cyclic (QC) codes of index t (see [15]) and A i 's are circulant matrices in the usual sense.…”

“…Whether such one-generator code families yield a CIS code (i.e. A i 's in (2) are all invertible) can be characterized by the polynomials a i (x) in (1). The following result is stated for QC codes in [6, Proposition 9.1].…”