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 2d−1≤ν<2d+1−1, then the binary code of the design contains a subcode that can be shortened to the binary Hamming codeH d of length 2d−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 2d, whered is as above.
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Communicated by R. Mullin
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Key, J.D., Sullivan, F.E. Codes of Steiner triple and quadruple systems. Des Codes Crypt 3, 117–125 (1993). https://doi.org/10.1007/BF01388410
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DOI: https://doi.org/10.1007/BF01388410