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Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 1/2015

01-04-2015

On relative constant-weight codes

Authors: Zihui Liu, Xin-Wen Wu

Published in: Designs, Codes and Cryptography | Issue 1/2015

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Abstract

In this paper, relative two-weight and three-weight codes are studied, which are both called relative constant-weight codes. A geometric approach is introduced to construct and characterize relative constant-weight codes, using the finite projective geometry. A sufficient and necessary condition is derived for linear codes to be relative constant-weight codes, based on the geometric approach. A family of infinite number of relative constant-weight codes are constructed, which includes dual Hamming codes and subcodes of punctured Reed–Muller codes as special instances. It is well known that determining all the minimal codewords is a hard problem for an arbitrary linear code. For relative constant-weight codes, minimal codewords are completely determined in this paper. Based on the above-mentioned results, applications of relative constant-weight codes to wire-tap channel of type II and secret sharing are discussed. A comparative study shows that relative constant-weight codes form a new family. They are not covered by the previously well-known three-weight codes or linear codes for which minimal codewords can be determined.
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Metadata
Title
On relative constant-weight codes
Authors
Zihui Liu
Xin-Wen Wu
Publication date
01-04-2015
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 1/2015
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-013-9896-2

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