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Binary codes of some strongly regular subgraphs of the McLaughlin graph

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Abstract

The article examines binary codes obtained from the row span of the adjacency matrices of some strongly regular graphs that occur as induced subgraphs of the McLaughlin graph, namely those with parameters (105, 32, 4, 12), (120, 42, 8, 18) and (253, 112, 36, 60). In addition we determine some primitive designs that are held by codewords of particular weights in the codes, and using the properties of the graphs and their geometry we provide a geometrical description of the nature of several classes of codewords. The codes with parameters [120, 100, 6]2 and [120, 101, 6]2 are near optimal as they are a distance 2 and 1 respectively more than the theoretical upper bound on the minimum distance for a code of the given length and dimension. Those with parameters [105, 87, 5]2 and [253, 231, 5]2 are a distance 1 less that the known recorded distance.

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Authors and Affiliations

  1. Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand

    Dimitri Leemans

  2. School of Mathematical Sciences, University of KwaZulu-Natal, Durban, 4041, South Africa

    B. G. Rodrigues

Authors
  1. Dimitri Leemans
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  2. B. G. Rodrigues
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Correspondence to B. G. Rodrigues.

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Communicated by D. Ghinelli.

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Leemans, D., Rodrigues, B.G. Binary codes of some strongly regular subgraphs of the McLaughlin graph. Des. Codes Cryptogr. 67, 93–109 (2013). https://doi.org/10.1007/s10623-011-9589-7

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  • DOI: https://doi.org/10.1007/s10623-011-9589-7

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