New non-binary quantum codes from constacyclic codes over <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $\end{document}</tex-math></inline-formula>

Abstract

Let R = Fq[u, v]/ u 2 − 1, v 2 − v, uv − vu be a finite non-chain ring, where q is an odd prime power and u 2 = 1, v 2 = v, uv = vu. In this paper, we construct new non-binary quantum codes from (α + βu + γv + δuv)constacyclic codes over R. We give the structure of (α + βu + γv + δuv)constacyclic codes over R and obtain self-orthogonal codes over Fq by Gray map. By using Calderbank-Shor-Steane (CSS) construction and Hermitian construction from dual-containing (α + βu + γv + δuv)-constacyclic codes over R, some new non-binary quantum codes are obtained.2010 Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 11T71. i=0 k i , d] linear code over F q , where k i is the dimension of linear code C i for i = 0, 1, 2, 3.

New quantum codes from constacyclic codes over the ring $ R_{k,m} $

“…Now, we review some important results on linear codes over R k,m . One can find the similar results in [7,19,30,36].…”

“…But, we define the map ψ as the multiplication of a vector by an invertible matrix of order 2 k . Such type of Gray maps one can also find in [13,29,30] with respect to their setup. One of the main advantages to choose such Gray maps, like ψ, is to enhance the code parameters (particularly, dimension, and minimum distance, etc.)…”

“…Also, (x 8 −δ i ) ≡ 0 (mod f i (x)f * i (x)), for i = 0, 1. Therefore, by Theorem 4.3, C ⊥ ⊆ C. Hence, by Theorem 4.5, there exists a quantum code [ [16,10,3]] 9 , which has the larger code rate than the known code [ [16,8,3]] 9 given by [30].…”

“…In this connection, many significant works have been reported in the literature which provides better quantum codes over the finite fields, see [14,15,16,26,32]. It is also observed that the linear (e.g., cyclic, constacyclic) codes over finite non-chain rings produced a huge amount of good quantum codes [1,2,8,11,13,19,12,27,29,30]. In 2015, Ashraf and Mohammad [1] studied quantum codes from cyclic codes over F p + vF p .…”

“…There are a lot of articles in which good quantum codes are obtained from the cyclic codes on different finite rings, see [11,13,19,23,31,33,32,34]. On the other side, recently, Gao and Wang [12], Li et al [27], Ma et al [29,30] considered the constacyclic codes over finite non-chain rings and obtained many new and better codes compare to the known codes. Based on the above studies, one can say that the constacyclic codes are a great resource to supply good quantum codes over finite rings.…”