New q-ary quantum MDS codes with distances bigger than $$\frac{q}{2}$$ q 2

Abstract

Constructions of quantum MDS codes have been studied by many authors. We refer to the table in page 1482 of [3] for known constructions. However there have been constructed only a few q-ary quantum MDS [[n, n−2d+2, d]] q codes with minimum distances d > q 2 for sparse lengths n > q + 1. In the case n = q 2 −1 m where m|q + 1 or m|q − 1 there are complete results. In the case n = q 2 −1 m while m|q 2 − 1 is not a factor of q − 1 or q + 1, there is no q-ary quantum MDS code with d > q 2 has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over F q 2 . Then we give some new q-ary quantum codes in this case. Moreover we present many new q-ary quantum MDS codes with lengths of the form w(q 2 −1) u and minimum distances d > q 2 .

Determining when a truncated generalised Reed-Solomon code is Hermitian self-orthogonal

“…There are many constructions of quantum MDS codes with d q + 1, mostly based on cyclic or constacyclic constructions and generalised Reed-Solomon codes. For example those contained in [4][5][6], [8,9], [11], [12][13][14][15], [17,18], [20][21][22] and [23][24][25].…”

Two Families of Entanglement-Assisted Quantum MDS Codes from Constacyclic Codes

“…Therefore, constructing QMDS codes and EAQMDS codes have became a central topic for quantum error-correction codes. Currently, many QMDS codes have been constructed by different methods [17][18][19][20][21][22][23][24][25]. In [5], the MDS conjecture shown that the length of maximal-distance-separable (MDS) code cannot exceed q 2 + 1.…”

Classical and Quantum Evaluation Codes at the Trace Roots

“…Although quantum codes were introducecd recently, the literature on this topic is very large. Most papers have addressed the study of quantum MDS, LDCP and BCH codes [31,11,1,25,27,23,34,26,22].…”