Construction of some new quantum MDS codes

Two new classes of quantum MDS codes

“…However, it is not an easy task to construct quantum MDS codes with length n > q + 1 and minimum distance d > q 2 + 1. Researchers have made a great effort to construct such quantum MDS codes via negacyclic codes (see [16]), constacyclic codes (see [10,13,15,17,21]), Pseudocyclic Codes (see [19]) and generalized Reed-Solomon codes (see [6,7,8,9,11,23]). We only list the known results for the constructions of q-ary quantum MDS codes with length n = q 2 or q 2 + 1 in Table 1.…”

Some New Constructions of Quantum MDS Codes

“…Applying the propagation rule (see Lemma 2) for Theorem 4 (i) and (ii), we immediately obtain the following corollaries which were given in [1] and [2], respectively. 2s (see Corollary 6).…”

“…Shi et al [2,Theorem 4.2] constructed a family of quantum MDS codes of length n = r q 2 −1 2s+1 , where r = 2t + 2 is even. For r = 2t + 1 odd, applying the propagation rule (see Lemma 2) for Theorem 3 (ii), we can immediately obtain the following result.…”

“…Let (r, s) =(2,2) in Theorem 3 (i). Then, when 5 | (q + 1), there exists a[[1 + 2 5 (q 2 − 1), 1 + 2 5 (q 2 − 1) − 2k, k + 1]] q quantum MDS code for any 1 ≤ k ≤ 3 5 (q + 1) − 1;(ii) Let (r, s) = (5, 3) in Theorem 3 (ii).…”

Some Classes of New Quantum MDS and Synchronizable Codes Constructed From Repeated-Root Cyclic Codes of Length 6ps

“…After that, [51] gave two new classes of qMDS codes from negacyclic codes in 2013. Recent years, many researchers worked on construction of qMDS code with minimum distance larger than q 2 + 1 (for examples, [19], [31], [32], [73], [84], [85]). An [[n, k]] QEC code encodes k logical qubits into n physical qubits.…”