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

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Abstract

The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474–1484, 2015) 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 \(d>\frac{q}{2}\) for sparse lengths \(n>q+1\). In the case \(n=\frac{q^2-1}{m}\) where \(m|q+1\) or \(m|q-1\) there are complete results. In the case \(n=\frac{q^2-1}{m}\) while \(m|q^2-1\) is neither a factor of \(q-1\) nor \(q+1\), no q-ary quantum MDS code with \(d> \frac{q}{2}\) has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over \(\mathbf{F}_{q^2}\). Then we give some new q-ary quantum codes in this case. Moreover many new q-ary quantum MDS codes with lengths of the form \(\frac{w(q^2-1)}{u}\) and minimum distances \(d > \frac{q}{2}\) are presented.

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Correspondence to Hao Chen.

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X. He was supported by NSFC Grant 61202007; L. Xu and H. Chen were supported by NSFC Grants 11371138 and 11531002.

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He, X., Xu, L. & Chen, H. New q-ary quantum MDS codes with distances bigger than \(\frac{q}{2}\) . Quantum Inf Process 15, 2745–2758 (2016). https://doi.org/10.1007/s11128-016-1311-2

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  • DOI: https://doi.org/10.1007/s11128-016-1311-2

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