Quantum Codes from the Cyclic Codes Over $$\mathbb {F}_{p}[v,w]/\langle v^{2}-1,w^{2}-1,vw-wv\rangle $$

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

“…Also, (x 11 − 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 [ [22,2,7]] 5 , which has the larger distance compare to the known code [ [22,2,5]] 5 given by [10,17]. Remark 1.…”

“…over the parameters obtained by the simple canonical Gray map. For example, using ψ we construct quantum code [ [22,2,7]] 5 in Example 4.6 whose minimum distance is larger than the quantum code [ [22,2,5]] 5 obtained in [17] under the usual canonical Gray map. Now, we present an example for k = 2 to understand the ring structure based on the set of pairwise orthogonal idempotent elements and Gray map discussed above.…”

“…Then the Gray image ψ(C) has the parameters [22,12,7]. Also, (x 11 − δ i ) ≡ 0 (mod f i (x)f * i (x)), for i = 0, 1.…”

“…Remark 1. In the above example, we have calculated that the Gray image ψ(C) is a [22,12,7] linear code over F 5 . Note that ψ(C) has length = 2 k n = 2 1 • 11 = 22 and dimension is equal to the sum of the dimensions of linear codes generated by polynomials f 0 (x) and f 1 (x)=6 + 6 = 12.…”

“…Note that for k = 2, m = 1 the constacyclic codes over R k,m are studied in [21]. Further, authors constructed quantum codes based on cyclic codes over R 2,1 in [22] and over R 3,1 in [19], respectively. Therefore, the present article is a continuation and generalization of our previous studies in the context of new quantum codes construction.…”