Codes of Steiner triple and quadruple systems
type: Item
instance of: scholarly article
title: Codes of Steiner triple and quadruple systems
zbMATH Open document ID: 0779.51004
DOI: 10.1007/BF01388410
author: Yanyan Li
published in: Designs, Codes and Cryptography
publication date: 1994-01-18T00:00:00Z
review text: Let \({\mathcal D} = ({\mathcal P},{\mathcal B})\) be an incidence structure with point set \(\mathcal P\) of order \(\nu\) and with block set \(\mathcal B\). The code \(C_ p({\mathcal D})\) (of length \(\nu\)) of \(\mathcal D\) over a prime field \(F_ p\) is the subspace of the space \(F^{\mathcal P}_ p\) of all functions from \(\mathcal P\) to \(F_ p\) that is spanned by incidence vectors of the blocks of \(\mathcal D\). If every block \(\mathcal D\) is incident with precisely \(k\) points and set of \(t\) distinct points are together incident with precisely \(\lambda\) blocks, then \(\mathcal D\) is said to be \(t-(\nu,k,\lambda)\) design. A Steiner triple (quadruple) system is any \(2(\nu,3,1)\) (resp. \(3- (\nu,4,1)\)) design. If \(C\) is a code of length \(\nu\) and every codeword of \(C\) has coordinate 0 at a particular coordinate position, then the code obtained from \(C\) by deleting this coordinate position is said to be a shortened code of \(C\). Let \(d\) to be an integer such that \(2^ d-1 \leq \nu < 2^{d+1} - 1\). The authors show that if \(\mathcal D\) is a Steiner triple system, then the binary code \(C_ 2({\mathcal D})\) of \(\mathcal D\) contains a subcode that can be shortened to the binary Hamming code \({\mathcal K}_ d\) of length \(2^ d- 1\). Similarly the binary code of any Steiner quadruple system on \(\nu + 1\) points contains a subcode that can be shortened to the Reed-Muller code \({\mathcal R}(d-2,d)\) of order \(d-2\) and length \(2^ d\), where \(d\) is as above. These results are analogous with a result obtained by \textit{J. Doyen, X. Hubaut} and \textit{M. Vandensavel} in Math. Z. 163, 251-259 (1978; Zbl 0373.05011).
Mathematics Subject Classification ID: 51E10
Mathematics Subject Classification ID: 51E22
Mathematics Subject Classification ID: 05B07
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 202952
zbMATH Keywords: Steiner triple system
zbMATH Keywords: binary code
zbMATH Keywords: Steiner quadruple system
reviewed by: Anatoli Kondrat'ev
MaRDI profile type: Publication
cites work: Q4004114
cites work: Ranks of incidence matrices of t-designs \(S_\lambda (t,t+1,v)\)
cites work: On generalized ReedMuller codes and their relatives
cites work: Q5184922
cites work: Q4146667
cites work: Q3663313
cites work: On projective and affine hyperplanes
cites work: Q3815311
full work available at URL: https://doi.org/10.1007/bf01388410
OpenAlex ID: W2023887414
Recommended article: Ternary codes of steiner triple systems
Recommended article: Steiner systems from binary codes
Recommended article: Extendable Steiner designs from finite geometries
Recommended article: Q4863444
Recommended article: On the binary codes with parameters of doubly-shortened 1-perfect codes
Recommended article: On 2-ranks of Steiner triple systems
Recommended article: Q4393341
Recommended article: 3-(2m, 2m–1, 2m–2–1) design for first order Reed-Muller code R(1,m)
Recommended article: There exist Steiner triple systems of order 15 that do not occur in a perfect binary one‐error‐correcting code
Recommended article: The construction of some bit and byte error control codes using partial Steiner systems
description: scientific article; zbMATH DE number 202952