Steiner triple systems of order 15 and their codes
This paper studies the binary linear codes generated by the \(15\times 35\) incidence matrices of the 80 Steiner triple systems of order 15 (STS(15)). The 80 codes of length 35 spanned by incidence vectors of the 15 points are all non-isomorphic. However, a pair of codes of length 15, generated by blocks, are isomorphic if and only if the corresponding incidence matrices have the same rank over GF(2). The paper includes data on the weight distribution, the automorphism groups, and the distribution of the STS(15)s within the codes. Although non-isomorphic STS(15)s define inequivalent codes of length 35, there are 54 codes that contain several non-isomorphic STS(15) among the codewords of weight 7, and any such code is generated by the STS(15) of maximum 2-rank. Of particular interest is the fact that the code of length 35 with the largest automorphism group (of order \(225 792\)) is not the code of the geometric design PG(3, 2) (of order \(20 160\)), but the code of the STS(15) number 16 in \textit{R. A. Mathon, K. T. Phelps} and \textit{A. Rosa} [Ars Comb. 15, 3-110 (1983; Zbl 0516.05010)], whose automorphism group has order 168.
scientific article; zbMATH DE number 1040412
scientific article