Constructions of A Large Class of Optimum Constant Weight Codes over F_2

A new method of constructing optimum constant weight codes over F_2 based on a generalized \((u, u+v)\) construction is presented. We present a new method of constructing superimposed code \(C_{(s_1,s_2,\cdots,s_I)}^{(h_1, h_2, \cdots, h_I)}\) bound. and presented a large class of optimum constant weight codes over F_2 that meet the bound due to Brouwer and Verhoeff, which will be referred to as BV . We present large classes of optimum constant weight codes over F_2 for \(k=2\) and \(k=3\) for \(n \leqq 128\). We also present optimum constant weight codes over F_2 that meet the BV bound for \(k=2,3,4,5\) and 6, for \(n \leqq 128\). The authors would like to present the following conjectures : \(C_{I}\): \(C_{(s_1)}^{(h_1)}\) presented in this paper yields the optimum constant weight codes for the code-length \(n=3h_1\), number of information symbols \(k=2\) and minimum distance \(d=2h_1\) for any positive integer \(h_1\). \(C_{II}\): \(C_{(s_1)}^{(h_1)}\) yields the optimum constant weight codes at \(n=7h_1, k=3\) and \(d=4h_1\) for any \(h_1\). \(C_{III}\): Code \(C_{(s_1,s_2,\cdots,s_I)}^{(h_1, h_2, \cdots, h_I)}\) yields the optimum constant weight codes of length \(n=2^{k+1}-2\), and minimum distance \(d=2^{k}\) for any number of information symbols \(k\geq 3\).