The invariant factors of some cyclic difference sets
type: Item
instance of: scholarly article
title: The invariant factors of some cyclic difference sets
zbMATH Open document ID: 1032.05013
DOI: 10.1016/S0097-3165(02)00023-7
author: David B. Chandler
author: Qing Xiang
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-03-26T00:00:00Z
review text: Difference sets with parameters \(\left(\frac{q^m-1}{q-1}, \frac{q^{m-1}-1}{q-1}, \frac{q^{m-2}-1}{q-1}\right)\), \(q\) prime power, are called difference sets with classical parameters. Recently, many new constructions of such difference sets with \(q=2\) have been found, and two (infinite) series for \(q=3\). One series are the HKM difference sets [see \textit{T. Helleseth} et al., Des. Codes Cryptography 23, 157-166 (2001; Zbl 1007.94013)]. The other series are the Lin difference sets (conjectured in the Ph.D. thesis of \textit{H. A. Lin} [From cyclic Hadamard difference sets to perfectly balanced sequences. University of Southern California, Los Angeles (1998)]). Both difference sets have the same \(3\)-rank. The authors compute the Smith normal form (which is a refinement of computing the \(p\)-rank), therefore showing that the designs corresponding to the difference sets are not isomorphic. The proof uses algebraic number theory, for instance Stickelberger's theorem, Gauss sums and \(p\)-adic valuations.
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 94B15
zbMATH DE Number: 1888442
zbMATH Keywords: Gauss sum
zbMATH Keywords: Singer difference set
reviewed by: Alexander Pott
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1888442