Hypergraphical codes arising from binary trades (Q1591636)

The subject of this paper concerns binary codes arising from incidence matrices of \(v\)-subsets vs \(k\)-subsets. To be precise, for positive integers \(v,k\), with \(v\geq k+2\), let \(P^v_k\) be the \(v\times\binom{v}{k}\) incidence matrix of a \(2-(v,k\binom{v-2}{k-2})\) design. Then the authors study the code \(C^v_k:=\text{ ker}(X\mapsto P^v_k X)\subseteq GF(2)^{\binom{v}{k}}\). Firstly, the parameters of the codes are computed: a direct application of \textit{R. M. Wilson}'s paper [Eur. J. Comb. 11, No. 6, 609-615 (1990; Zbl 0747.05016)] shows that the dimension of \(C^v_k\) is \(\binom{v}{k}-v\) if \(k\) is odd, and \(\binom{v}{k}-v+1\) otherwise. On the other hand, the minimum distance of \(C^v_k\) is \(4\) whenever \(k\) is odd or \(k\) even and \(v<3k/2\); the minimum distance is \(3\) otherwise. As a matter of fact, some of these codes are optimal. Secondly, the authors compute the weight enumerator polynomial of \(C^v_k\). Their approach is based on \textit{D. Jungnickel} and \textit{S. A. Vanstone}'s ideas [Arch. Math. 65, 461-464 (1995; Zbl 0855.05069)] and they have to generalize the so-called bond spaces and potential differences of graphs.