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

AbstractFork 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.