A new construction of two-, three- and few-weight codes via our GU codes and their applications

Linear codes with few weights have applications in cryptography, association schemes, designs, strongly regular graphs, finite group theory, finite geometries, and secret sharing schemes, among other disciplines. Two-weight linear codes are particularly interesting because they are closely related t...

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Published in:Applicable algebra in engineering, communication and computing Vol. 33; no. 6; pp. 629 - 647
Main Authors: Eddie, Arrieta A, Janwa, Heeralal
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2022
Springer Nature B.V
Subjects:
Algorithms
Artificial Intelligence
Codes
Computer Hardware
Computer Science
Cryptography
Graphs
Group theory
Linear codes
Original Paper
Permutations
Symbolic and Algebraic Manipulation
Theory of Computation
Simplex code
Uniformly packed code
05
51E20
Antipodal linear codes
51EXX
20B10
Two- and three-weight codes
Optimal additive code
Quasi-perfect linear code
94
05B25
20B05
05B05
ISSN:0938-1279, 1432-0622
Online Access:Get full text
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Summary:Linear codes with few weights have applications in cryptography, association schemes, designs, strongly regular graphs, finite group theory, finite geometries, and secret sharing schemes, among other disciplines. Two-weight linear codes are particularly interesting because they are closely related to objects in different areas of mathematics such as strongly regular graphs, 3-rank permutation groups, ovals, and arcs. There exist techniques to construct linear codes with few weights, for example, the systematic exposition by Calderbank and Kantor (Bull Lond Math Soc 18(2):97–122, 1986 ). Ding et al., (World Sci, pp 119–124, 2008 ) and (IEEE Trans Inf Theory 61(11):5835–5842, 2015 ) constructed few-weight codes using the trace function and Tonchev et al. (Algorithms, 12(8), 2019 ) generalized Ding’s construction. In this paper, we present an elementary way to get two- and three-weight codes from simplex codes and antipodal linear codes. An interesting application is the construction of uniformly packed linear codes from two-weight codes and quaternary quasi-perfect linear codes from three-weight codes.
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ISSN:0938-1279
1432-0622
DOI:10.1007/s00200-022-00561-8