Codes of Steiner triple and quadruple systems

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 points, and if the integerd is such that 2 ^d –1<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 +1 points contains a subcode that can be shortened to the Reed-Muller code (d–2,d) of orderd–2 and length 2 ^d , whered is as above.