Skew-constacyclic codes over $\mathcal R$

Abstract

Let p be an odd prime and q = p m , where m is a positive integer. We study the Θ t -cyclic and (Θ t , λ )-cyclic code over a finite commutative non-chain ring R = F q [u, v, w]/ u 2 = u, v 2 = v, w 2 = 1, uv = vu = 0, uw = wu, wv = vw , where λ is a unit in R.

QUANTUM CODES FROM CYCLIC CODES OVER THE F_q+uF_q+vF_q+wF_q+uvF_q

“…In [7], the commutative ring with introduced, where is a finite field with elements, and is an odd prime. The other results have appeared in [7]. , where denotes the ordinary Hamming weight for codes over .…”

“…Lemma 1. [7] The Gray map is a distance-preserving map or isometry from (Lee distance) to (Hamming distance) and it is also -linear.…”

“…If are the generator matrices of -ary linear codes , , respectively, then the generator matrix of is and the generator matrix of is Theorem 4. [7] Let be a linear code of length over . Then .…”

“…Theorem 6. [7] Let be a linear code of length over . Then is a cyclic code if and only if , are cyclic codes over .…”

Quantum Codes from (1-2w-2uv)-Constacyclic Codes over the F_q+uF_q+vF_q+wF_q+uvF_q

“…In this paper, motivated by the previous works (Gao and Wang, 2018), (Li et al, 2018), (Islam and Prakash, 2020), (Mohan and Durairajan, 2020), we study the quantum codes which are obtained from (1 − 2𝑤 − 2𝑢𝑣)-constacyclic codes over the finite ring 𝔽 𝑞 + 𝑢𝔽 𝑞 + 𝑣𝔽 𝑞 + 𝑤𝔽 𝑞 + 𝑢𝑣𝔽 𝑞 .…”