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Completely reducible super-simple designs with block size five and index two

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Abstract

Complete reducible super-simple (CRSS) designs are closely related to \(q\)-ary constant weight codes. A \((v,k,\lambda )\)-CRSS design is just an optimal \((v,2(k-1),k)_{\lambda +1}\) code. In this paper, we mainly investigate the existence of a \((v,5,2)\)-CRSS design and show that such a design exists if and only if \(v\equiv 1,5\pmod {20}\) and \(v\ge 21\), except possibly when \(v = 25\). Using this result, we determine the maximum size of an \((n,8,5)_3\) code for all \(n\equiv 0,1,4,5 \pmod {20}\) with the only length \(n=25\) unsettled. In addition, we also construct super-simple \((v,5,3)\)-BIBDs for \(v=45,65\).

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Acknowledgments

Research supported by the National Natural Science Foundation of China under Grant No. 61171198 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LZ13A010001.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou , 310027, Zhejiang, China

    Hengjia Wei & Hui Zhang

  2. School of Mathematical Sciences, Capital Normal University, Beijing , 100048, China

    Gennian Ge

  3. Beijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing , 100048, China

    Gennian Ge

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  1. Hengjia Wei
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  2. Hui Zhang
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Correspondence to Gennian Ge.

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Communicated by L. Teirlinck.

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Wei, H., Zhang, H. & Ge, G. Completely reducible super-simple designs with block size five and index two. Des. Codes Cryptogr. 76, 589–600 (2015). https://doi.org/10.1007/s10623-014-9979-8

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