Binary Codes from Graphs on Triples and Permutation Decoding

Abstract

We show that permutation decoding can be used, and give explicit PD-sets in the symmetric group, for some of the binary codes obtained from the adjacency matrices of the graphs on \(\binom{n}{3}\) vertices, for \(n \geq 7\), with adjacency defined by the vertices as \(3\)-sets being adjacent if they have zero, one or two elements in common, respectively.