Ternary codes of steiner triple systems

Abstract

The code over a finite field Fq of a design 𝒟 is the space spanned by the incidence vectors of the blocks. It is shown here that if 𝒟 is a Steiner triple system on v points, and if the integer d is such that 3d ≤ v < 3d+1, then the ternary code C of 𝒟 contains a subcode that can be shortened to the ternary generalized Reed-Muller code ℛF3(2(d − 1),d) of length 3d. If v = 3d and d ≥ 2, then C⟂ ⊆ ℛF3(1,d)⊆ ℛ F3(2(d − 1),d) ⊆ C. © 1994 John Wiley & Sons, Inc.