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Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes

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Abstract

It has become common knowledge that constructing q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\) is significantly more difficult than constructing those with minimum distance less than or equal to \(q/2+1\). Despite of various constructions of q-ary quantum MDS codes, all known q-ary quantum MDS codes have minimum distance bounded by \(q/2+1\) except for some lengths. The purpose of the current paper is to provide some new q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\). In this paper, we provide several classes of quantum MDS codes with minimum distance bigger than \(q/2+1\). For instance, some examples in these classes include q-ary \([n,n-2k, k+1]\)-quantum MDS codes for cases: (i) \(q\equiv -1\bmod {5}, n=(q^2+4)/5\) and \(1\le k\le (3q-2)/5\); (ii) \(q\equiv -1\bmod {7}, n=(q^2+6)/7\) and \(1\le k\le (4q-3)/7\); (iii) \(2|q, q\equiv -1\bmod {3}, n=2(q^2-1)/3\) and \(1\le k\le (2q-1)/3\); and (iv) \(2|q, q\equiv -1\bmod {5}, n=2(q^2-1)/5\) and \(1\le k\le (3q-2)/5\).

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Acknowledgments

The authors would like to thank the referees for many helpful comments. The research work of the first author is supported in part by Shanghai Sailing Program under the Grant 15YF1401200 and by the National Natural Science Foundation of China under Grant 11501117. The research work of the second author and third author are supported in part by the National Natural Science Foundation of China under Grant 61672166, in part by the Shanghai Excellent Academic Leaders under Grant 16XD1400200, and in part by the Shanghai Innovation Plan of Science and Technology under Grant 16JC1402700.

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Authors and Affiliations

  1. School of Computer Science, Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai, 200433, China

    Lingfei Jin, Haibin Kan & Jie Wen

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  1. Lingfei Jin
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  2. Haibin Kan
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  3. Jie Wen
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Correspondence to Lingfei Jin.

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Communicated by C. Mitchell.

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Jin, L., Kan, H. & Wen, J. Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes. Des. Codes Cryptogr. 84, 463–471 (2017). https://doi.org/10.1007/s10623-016-0281-9

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  • DOI: https://doi.org/10.1007/s10623-016-0281-9

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