PD-sets for codes related to flag-transitive symmetric designs

Abstract

‎For any prime $p$ let $C_p(G)$ be the $p$-ary code spanned by the rows of the incidence matrix $G$ of a graph $Gamma$‎. ‎Let $Gamma$ be the incidence graph of a flag-transitive symmetric design $D$‎. ‎We show that any flag-transitive‎ ‎automorphism group of $D$ can be used as a PD-set for full error correction for the linear code $C_p(G)$‎ ‎(with any information set)‎. ‎It follows that such codes derived from flag-transitive symmetric designs can be‎ ‎decoded using permutation decoding‎. ‎In that way to each flag-transitive symmetric $(v‎, ‎k‎, ‎lambda)$ design we associate a linear code of length $vk$ that is‎ ‎permutation decodable‎. ‎PD-sets obtained in the described way are usually of large cardinality‎. ‎By studying codes arising from some flag-transitive symmetric designs we show that smaller PD-sets can be found for‎ ‎specific information sets‎.