On Bonisoli’s theorem and the block codes of Steiner triple systems

Abstract

A famous result of Bonisoli characterizes the equidistant linear codes over $${\mathrm{GF}}(q)$$ (up to monomial equivalence) as replications of some q-ary simplex code, possibly with added 0-coordinates. We first prove a variation of this theorem which characterizes the replications of first order generalized Reed–Muller codes among the two-weight linear codes. In the second part of this paper, we use Bonisoli’s theorem and our variation to study the linear block codes of Steiner triple systems, which can only be non-trivial in the binary and ternary case. Assmus proved that the block by point incidence matrices of all Steiner triple systems on v points which have the same 2-rank generate equivalent binary codes and gave an explicit description of a generator matrix for such a code. We provide an alternative, considerably simpler, proof for these results by constructing parity check matrices for the binary codes spanned by the incidence matrix of a Steiner triple system instead, and we also obtain analogues for the ternary case. Moreover, we give simple alternative coding theoretical proofs for the lower bounds of Doyen, Hubaut and Vandensavel on the 2- and 3-ranks of Steiner triple systems.