Generalizing binary quadratic residue codes to higher power residues over larger fields
type: Item
instance of: scholarly article
title: Generalizing binary quadratic residue codes to higher power residues over larger fields
zbMATH Open document ID: 1167.94006
author: P. Charters
published in: Finite Fields and their Applications
publication date: 2009-05-19T00:00:00Z
full work available at URL: http://hdl.handle.net/2152/11667
review text: The binary quadratic residue codes are well-known codes in coding theory. Their construction starts from \(p\equiv \pm1\pmod{8}\) so that 2 is a quadratic residue modulo \(p\). This article investigates a generalization of these quadratic residue codes. Let \(p\) and \(q\) be two distinct prime numbers such that \(q|(p-1)\) and such that \(q\) is a \(q\)-th power residue modulo \(p\). Then there exists an element \(\beta\in\mathbb{F}_p\) such that \(\beta^q\equiv q \pmod{p}\), and it is possible to divide \(\mathbb{F}_p\) into \(q\) cosets. These cosets are the basis to define the \(q\)-th power residue codes. The author presents generating polynomials for these codes, and defines a new notion corresponding to the binary concept of an idempotent. The construction of these \(q\)-th power residue codes is also briefly explained in Section 15.2 of \textit{E. W. Berlekamp} [Algebraic Coding Theory (McGraw-Hill, New York, NY) (1968; Zbl 0988.94521)]. Using different techniques from the book of Berlekamp, the author presents a lower bound on the codeword weight of the dual of these \(q\)-th power residue codes. This lower bound then leads to a lower bound on the weight of the codewords of the \(q\)-th power residue codes.
Mathematics Subject Classification ID: 94B15
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 5554720
zbMATH Keywords: quadratic residue codes
zbMATH Keywords: coding theory
zbMATH Keywords: idempotents
reviewed by: Leo Storme
MaRDI profile type: Publication
OpenAlex ID: W1973534971
cites work: Algebraic coding theory
cites work: Legendre sums and codes related to QR codes
cites work: Double Circulant Quadratic Residue Codes
cites work: Q4146667
cites work: A generalisation of quadratic residue codes to the case of cubic and biquadratic residues
DOI: 10.1016/J.FFA.2009.02.001
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description: scientific article; zbMATH DE number 5554720