Strongly regular graphs from reducible cyclic codes
type: Item
instance of: scholarly article
title: Strongly regular graphs from reducible cyclic codes
zbMATH Open document ID: 1501.94113
author: Minjia Shi
author: Tor Helleseth
author: Patrick Solé
published in: Journal of Algebraic Combinatorics
publication date: 2022-03-15T00:00:00Z
review text: ``Since the seminal paper of \textit{Ph. Delsarte} [Discrete Math. 3, 47--64 (1972; Zbl 0245.94010)] two-weight codes have been studied in symbiosis with combinatorial objects like strongly regular graphs (SRGs) [\textit{A. E. Brouwer} et al., Distance-regular graphs. Berlin etc.: Springer-Verlag (1989; Zbl 0747.05073); \textit{E. Byrne} et al., Des. Codes Cryptography 48, No. 1, 1--16 (2008; Zbl 1184.94266); \textit{E. Byrne} et al., Des. Codes Cryptography 48, No. 1, 1--16 (2008; Zbl 1184.94266); \textit{Ph. Delsarte}, Discrete Math. 3, 47--64 (1972; Zbl 0245.94010)] or geometrical structures like caps in projective spaces [\textit{R. Calderbank} and \textit{W. M. Kantor}, Bull. Lond. Math. Soc. 18, 97--122 (1986; Zbl 0582.94019)]. An important class of codes for construction of two-weight codes is that of irreducible cyclic codes [\textit{L. D. Baumert} and \textit{R. J. McEliece}, Inf. Control 20, 158--175 (1972; Zbl 0239.94007); \textit{G. Vega}, Finite Fields Appl. 33, 1--13 (2015; Zbl 1368.11134)]. In particular, it is conjectured that all two-weight projective irreducible cyclic codes are known [\textit{B. Schmidt} and \textit{C. White}, Finite Fields Appl. 8, No. 1, 1--17 (2002; Zbl 1023.94016); \textit{G. Vega}, Finite Fields Appl. 33, 1--13 (2015; Zbl 1368.11134)]. More recently, reducible cyclic codes have been used as a source of construction of two-weight codes. The connection between SRG's and two-weight codes over finite fields was extended recently to two-weight codes over rings for the homogeneous weight [\textit{E. Byrne} et al., Des. Codes Cryptography 48, No. 1, 1--16 (2008; Zbl 1184.94266); Finite Fields Appl. 18, No. 4, 711--727 (2012; Zbl 1245.05135)].'' Let \(p\) be an odd prime. A code \(C\subseteq \mathbb{Z}_{p^m}\) is called linear if \(C\) is a submodule of \(\mathbb{Z}_{p^m}.\) As usual, a code \(C\) is called cyclic if \(C\) is linear and invariant under the cyclic shift. Let \(C=\langle g(x) \rangle\) be a cyclic code over \(\mathbb{Z…
reviewed by: Djoko Suprijanto
Mathematics Subject Classification ID: 94B15
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 7490055
zbMATH Keywords: 2-weight codes
zbMATH Keywords: reducible cyclic codes
zbMATH Keywords: strongly regular graphs
MaRDI profile type: Publication
full work available at URL: s10801-020-01006-6
OpenAlex ID: W3151572268
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DOI: 10.1007/S10801-020-01006-6
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description: scientific article; zbMATH DE number 7490055