Equidistant families of sets
type: Item
instance of: scholarly article
title: Equidistant families of sets
zbMATH Open document ID: 0846.05007
DOI: 10.1016/0024-3795(95)00103-X
published in: Linear Algebra and its Applications
publication date: 1996-09-29T00:00:00Z
review text: This is a very interesting paper on the relation between large equidistant codes and combinatorial designs. The authors use the terminology of set theory. In this review, we will use \({\mathbf a}\) for the binary word of length \(v\) which has 1's in the positions corresponding to the subset \(A\) of \([v]:= \{1, 2,\dots, v\}\). In this terminology \({\mathbf a}+ {\mathbf b}\) corresponds to the symmetric difference \(A \Delta B\). Two families of subsets are called equivalent if the corresponding binary codes are translates. Of the two main results, the first concerns an equidistant code \(C\) of length \(v\) \((v\geq 3)\), distance \(d\) and cardinality \(v\). It states that if \(v= 2d\), then \(C\) is equivalent to a code consisting of the all one vector and the \(v- 1\) blocks of a Hadamard 3-design with no two blocks being complementary, and if \(v\neq 2d\), then \(C\) is equivalent to the block-set of a symmetric 2-design. The other main result extends a theorem of \textit{D. R. Stinson} and \textit{G. H. J. van Rees} [The equivalence of certain equidistant binary codes and symmetric BIBDs, Combinatorica 4, 357-362 (1984; Zbl 0556.05010)]. They proved that an equidistant binary code with wordlength \(v= 4k+ 1\) and distance \(d= 2k\) exists if and only if \(k= {1\over 2} (u^2+ u)\) and there exists a 2-\((2u^2+ 2u+ 1, u^2, {1\over 2}(u^2- u))\) design. The present authors give an alternative proof of this result and they extend it to the cases \(v\equiv 0, 2, 3\pmod 4\). In each of these cases the code is equivalent to some design. The proofs are based on the method of linear polynomials. For an equidistant code of length \(v\) and distance \(d\), associate with the set \(A\) (corresponding to codeword \({\mathbf a})\) the function \[ f_A(x_1, x_2,\dots, x_v):= \sum_{i\not\in A} x_i- \sum_{i\in A} x_i+ |A|- d. \] For any subset \(X\) of \([v]\) we have \[ f_A(X)= |A \Delta X|- d.\tag{\(*\)} \] The relation \((*)\) easily leads to the result of \textit{P. Dels…
reviewed by: Jack van Lint
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 94B25
zbMATH DE Number: 803738
zbMATH Keywords: Hadamard design
zbMATH Keywords: equidistant codes
zbMATH Keywords: combinatorial designs
zbMATH Keywords: binary codes
zbMATH Keywords: linear polynomials
zbMATH Keywords: codewords
author: Yury J. Ionin
author: M. S. Shrikhande
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 803738