The equivalence of certain equidistant binary codes and symmetric bibds

We study equidistant codes of length 4k + 1 having (constant) weight 2k, and (constant) distance 2k between codewords. The maximum number of codewords is 4k; this can be attained if and only ifk = (u ² +u)/2 (for some integeru) and there exists a ((2u ² + 2u + 1,u ², (u ² −u)/2) — SBIBD. Also, one can construct such a code, with 4k − 1 codewords, from a (4k − 1, 2k − 1,k − 1) — SBIBD. Supported, in part by NSERC grants U0217 (D. R. Stinson), A3558 (G. H. J. van Rees).