Orthogonal vectors in then-dimensional cube and codes with missing distances

Abstract

Fork a positive integer letm(4k) denote the maximum number of ±1-vectors of length 4k so that no two are orthogonal. Equivalently,m(4k) is the maximal number of codewords in a code of length 4k over an alphabet of size two, such that no two codewords have Hamming distance 2k. It is proved thatm(4k)=4\(\sum\limits_{0 \leqq i< k} {\left( {\begin{array}{*{20}c} {4k - 1} \\ i \\ \end{array} } \right)} \) ifk is the power of an odd prime.

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Frankl, P. Orthogonal vectors in then-dimensional cube and codes with missing distances. Combinatorica 6, 279–285 (1986). https://doi.org/10.1007/BF02579389

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AMS subject classification (1980)

  • 05 C 35