$ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes

Abstract

We study mixed alphabet cyclic and constacyclic codes over the two alphabets Z 4 , the ring of integers modulo 4, and its quadratic extension Z 4 [u] = Z 4 + uZ 4 , u 2 = 0. Their generator polynomials and minimal spanning sets are obtained. Further, under new Gray maps, we find cyclic, quasi-cyclic codes over Z 4 as the Gray images of both λ-constacyclic and skew λ-constacyclic codes over Z 4 [u]. Moreover, it is proved that the Gray images of Z 4 Z 4 [u]-additive constacyclic and skew Z 4 Z 4 [u]-additive constacyclic codes are generalized quasi-cyclic codes over Z 4 . Finally, several new quaternary linear codes are obtained from these cyclic and constacyclic codes.

ℤ₂ℤ₄-Additive Quasi-Cyclic Codes

“…Many generalizations are possible, by considering as alphabet pair a ring and one of its extensions; for instance Z 4 and Z 4 [u] with u 2 = 0, [15], or other pairs of rings [3], [4], [5], [9]. In another direction, the concept of quasi-cyclic codes could be extended to so-called generalized quasi-cyclic codes or quasi-abelian codes, or quasi-polycyclic codes [2].…”

“…The latter class was shown recently to be asymptotically good [12], [27]. Many other pairs of alphabets are possible [3], [4], [5], [9], [15], [24], but in the present paper, for simplicity's sake, we will focus on Z 2 and Z 4 . A natural generalization of cyclic codes is the class of quasicyclic codes, which has been studied in particular by using a decomposition of the alphabet into local rings and of the codes into constituent codes [16], [17], [25], by the Chinese Remainder Theorem (CRT).…”

On $ \mathbb{Z}_4\mathbb{Z}_4[u^3] $-additive constacyclic codes

“…Further, mixed alphabets additive constacyclic codes are extensively studied in [19,20,17]. Again, towards the generalization of these codes over mixed alphabets Z 2 r Z 2 s [u] where u 2 = 0, the reference [17] studied Z 4 Z 4 [u]-additive cyclic and constacyclic codes for s = r = 2.…”