Codes of Steiner triple and quadruple systems

Abstract

The code over a finite fieldF q of orderq of a design is the subspace spanned by the incidence vectors of the blocks. It is shown here that if the design is a Steiner triple system on ν points, and if the integerd is such that 2 d −1≤ν<2 d+1−1, then the binary code of the design contains a subcode that can be shortened to the binary Hamming codeH d of length 2 d −1. Similarly the binary code of any Steiner quadruple system on ν+1 points contains a subcode that can be shortened to the Reed-Muller code ℜ(d−2,d) of orderd−2 and length 2 d , whered is as above.