A note on skew constacyclic codes over 𝔽q + u𝔽q + v𝔽q

Abstract

In this paper, the skew constacyclic codes over finite non-chain ring [Formula: see text], where [Formula: see text], [Formula: see text] is an odd prime and [Formula: see text], are studied. We show that the Gray image of a skew [Formula: see text]-constacyclic code of length [Formula: see text] over [Formula: see text] is a skew quasi-twisted code of length [Formula: see text] over [Formula: see text] of index 3. Further, the structural properties of skew constacyclic codes over [Formula: see text] are obtained by the decomposition method.

Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

“…Throughout, we fix R = F q +uF q +vF q , u 2 = u, v 2 = v, uv = vu = 0. Now, following [3,12], we recall that R is a semi-local non-chain ring with three maximal ideals 1 − u , 1 − v and u + v . Again, by applying the Chinese Remainder Theorem (CRT) decomposition, we can write R 3 units and q 3 − (q − 1) 3 nonunit elements where the set of units is calculated by the fact that r is a unit in R if and only if r 1 , r 2 , r 3 are non-zero in F q .…”

“…), for all a, b, c ∈ F q . In addition, we consider another Gray map φ 2 : R → F 3 q defined in [12] by φ 2 (a + ub + vc) = (a, a + b, a + c), for all a, b, c ∈ F q . It is evident to check that φ i is an F q -linear bijective map and can be naturally extended over R n .…”

“…Finally, by taking Gray images of such codes we show that both families are good in terms of distance bounds (Theorem 2). It is worth mentioning that the ring R has an important interest and several classes of codes were considered over it [3,12,13] in the literature.…”

Skew Constacyclic Codes Over the Ring

“…Furthermore, T. Yao et al also studied the decomposition of skew cyclic code over the finite chain ring H = F q + uF q + vF q + uvF q with u 2 = u, v 2 = v, uv = vu, q = p m , with p odd prime ( according to Frobenius automorphism)( [10]). Later on, H. Islam et al introduced the Frobenius automorphism over H ( [11]). Similarly, A. Dertli et al defined the Gray map from…”

A note on constacyclic and skew constacyclic codes over the ring Zp[u, v]/hu 2 − u, v2 − v, uv − vui

“…Recently, researchers [11,13,14] have studied constacyclic codes over the extension ring Z 4 + uZ 4 of Z 4 and obtained several new linear codes over Z 4 from Gray images of these codes. For more works on the topic, we refer [3,4,[6][7][8].…”