Designs with the symmetric difference property on 64 points and their groups
type: Item
instance of: scholarly article
title: Designs with the symmetric difference property on 64 points and their groups
zbMATH Open document ID: 0813.05010
DOI: 10.1016/0097-3165(94)90002-7
author: Chris Parker
author: Edward Spence
author: Vladimir D. Tonchev
published in: Journal of Combinatorial Theory. Series A
publication date: 1995-05-17T00:00:00Z
review text: A symmetric 2-design has the symmetric difference property, or is an SDP design, if the symmetric difference of any three blocks is either a block or the complement of a block. The parameters \((v,k,\lambda)\) of a symmetric SDP design are of the form \[ v= 2^{2m},\quad k= 2^{2m-1}-2^{m-1},\quad \lambda= 2^{2m-2}-2^{m-1}. \] The symmetric SDP designs were characterized by \textit{J. F. Dillon} and \textit{J. R. Schatz} [Block designs with the symmetric difference property, Proc. NSA Math. Sci. Meet., 159-164 (1987)] as designs formed by the minimum weight vectors in a binary code spanned by a first-order Reed-Muller code and the incidence vector of a bent function. There are precisely four nonisomorphic SDP \(2\)-\((64,28,12)\) designs. In this paper the authors report the results of the computation of the automorphism groups of these four designs as well as the groups of their derived and residual designs. One interesting fact is that all four designs can be obtained from a difference set. As an application, the authors are able to enumerate the binary self-complementary \((28,7,12)\) and \((36,7,16)\) codes up to equivalence.
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 94B65
zbMATH DE Number: 617607
zbMATH Keywords: symmetric design
zbMATH Keywords: symmetric difference property
zbMATH Keywords: SDP design
zbMATH Keywords: symmetric SDP design
zbMATH Keywords: binary code
zbMATH Keywords: automorphism groups
zbMATH Keywords: residual designs
zbMATH Keywords: difference set
reviewed by: J. H. Dinitz
MaRDI profile type: Publication
full work available at URL: 0097-3165(94)90002-7
OpenAlex ID: W1990212237
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description: scientific article; zbMATH DE number 617607