The nonexistence of code words of weight 16 in a projective plane of order 10
type: Item
instance of: scholarly article
title: The nonexistence of code words of weight 16 in a projective plane of order 10
zbMATH Open document ID: 0591.51015
DOI: 10.1016/0097-3165(86)90091-9
author: S. Swiercz
author: Clement Lam
author: L. H. Thiel
published in: Journal of Combinatorial Theory. Series A
publication date: 1986-01-01T00:00:00Z
review text: The well known unsolved problem of the existence of a projective plane of order 10 is being studied by means of the binary error correction code generated by the rows of the incidence matrix of the plane, assuming that the plane does exist. \textit{F. J. MacWilliams}, \textit{N. J. A. Sloane} and \textit{J. G. Thompson} [J. Comb. Theory, Ser. A 14, 66-78 (1973; Zbl 0251.05020)] found that the weight enumerator of this code is determined by the number of vectors of weights 12, 15 and 16. In later work, it was found that no code words of weights 12 and 15 exist in the code. The search for code words of weight 16 can be divided into 6 cases. \textit{J. L. Carter} [''On the existence of a projective plane of order ten'', Ph. D. Thesis, Univ. California, Berkeley (1974)] finished the search for 4 of these cases and part of the fifth. In the paper under review, the study of the remaining cases is completed, and the weight enumerator of the code is determined. No plane of order 10 was found. The weight enumerator shows that a weight 19 code word must exist if a plane exists. The authors thus recommend a search starting from a 19-point configuration.
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 94B25
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 3949156
zbMATH Keywords: weight of code word
zbMATH Keywords: projective plane of order 10
zbMATH Keywords: binary error correction code
zbMATH Keywords: weight enumerator
MaRDI profile type: Publication
full work available at URL: 0097-3165(86)90091-9
OpenAlex ID: W2070832461
cites work: Configurations in a Plane of Order Ten
cites work: Estimating the Efficiency of Backtrack Programs
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cites work: Q3029691
cites work: On the existence of a projective plane of order 10
cites work: Weight enumerators of self-orthogonal codes
cites work: On vectors of weight 16 in the code of a projective plane of order 10
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description: scientific article; zbMATH DE number 3949156