Let \(A(n,2\delta,w)\) denote the maximum number of codewords in any binary code of length \(n\), constant weight \(w\), and Hamming distance \(2\delta\). Several lower bounds for \(A(n,2\delta,w)\) are given. For \(w\) and \(\delta\) fixed, \(A(n,2\delta,w) \geq n^{w-\delta +l}/w!\) and \(A(n,4,w)\sim n^{w-1}/w!\) as \(n\to\infty\). In most cases these are better than the “Gilbert bound”. Revised tables of \(A(n,2\delta,w)\) are given in the range \(n\leq 24\) and \(\delta \leq 5\).
Some sequels to this paper are the papers by the authors [Combinatorics, graph theory and computing, Proc. West Coast Conf., Arcata/Calif. 1979, 25–40 (1980; Zbl 0452.05036)], [SIAM J. Algebraic Discrete Methods 1, 382–404 (1980; Zbl 0499.05049)] and by A. E. Brouwer, P. Delsarte and P. Piret [IEEE Trans. Inf. Theory 26, 742–743 (1980; Zbl 0466.94021)].