Generalized perfect arrays and Menon difference sets
type: Item
instance of: scholarly article
title: Generalized perfect arrays and Menon difference sets
zbMATH Open document ID: 0767.05030
DOI: 10.1007/BF00124209
author: Jonathan Jedwab
published in: Designs, Codes and Cryptography
publication date: 1993-01-16T00:00:00Z
review text: Given an \(s_ 1\times\cdots\times s_ r\) integer-valued array \(A\) and a \((0,1)\) vector \({\mathbf z}=(z_ 1,\dots,z_ r)\), form the array \(A'\) from \(A\) by recursively adjoining a negative copy of the current array for each dimension \(i\) where \(z_ i=1\). \(A\) is a generalized perfect array type \({\mathbf z}\) if all periodic autocorrelation coefficients of \(A'\) are zero, except for shifts \((u_ 1,\dots,u_ r)\) where \(u_ i\equiv 0\pmod{s_ i}\) for all \(i\). The array is perfect if \({\mathbf z}=(0,\dots,0)\) and binary if the array elements are all \(\pm 1\). A non- trivial perfect binary array (PBA) is equivalent to a Menon difference set in an Abelian group. Using only elementary techniques, we prove various construction theorems for generalized perfect arrays and establish conditions on their existence. We show that a generalized PBA whose type is not \((0,\dots,0)\) is equivalent to a relative difference set in an Abelian factor group. We recursively construct several infinite families of generalized PBAs, and deduce nonexistence results for generalized PBAs whose type is not \((0,\dots,0)\) from well-known nonexistence results for PBAs. A central result is that a PBA with \(2^{2y} 3^{2u}\) elements and no dimension divisible by 9 exists if and only if no dimension is divisible by \(2^{y+2}\). The results presented here include and enlarge the set of sizes of all previously known generalized PBAs.
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 95047
zbMATH Keywords: periodic autocorrelation
zbMATH Keywords: perfect binary array
zbMATH Keywords: Menon difference set
zbMATH Keywords: Abelian group
zbMATH Keywords: generalized perfect arrays
zbMATH Keywords: relative difference set
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 95047