Steiner triple systems of order 15 and their codes
type: Item
instance of: scholarly article
title: Steiner triple systems of order 15 and their codes
zbMATH Open document ID: 0943.05016
DOI: 10.1016/S0378-3758(96)00071-7
author: Robert S. Weishaar
author: Vladimir D. Tonchev
published in: Journal of Statistical Planning and Inference
publication date: 2000-08-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05B07
Mathematics Subject Classification ID: 94B05
Mathematics Subject Classification ID: 51E10
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1040412
zbMATH Keywords: Steiner triple system
zbMATH Keywords: binary code
zbMATH Keywords: automorphism group
zbMATH Keywords: finite geometry
reviewed by: Elizabeth J. Billington
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1040412