TY - JOUR
T1 - On relative constant-weight codes
AU - Liu, Zihui
AU - Wu, Xin Wen
N1 - Publisher Copyright:
© 2013, Springer Science+Business Media New York.
PY - 2015/4
Y1 - 2015/4
N2 - 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.
AB - 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.
KW - Finite projective geometry
KW - Minimal codeword
KW - Relative three-weight code
KW - Relative two-weight code
KW - Secret sharing
KW - Wire-tap channel of type II
UR - http://www.scopus.com/inward/record.url?scp=84925290985&partnerID=8YFLogxK
U2 - 10.1007/s10623-013-9896-2
DO - 10.1007/s10623-013-9896-2
M3 - Article
AN - SCOPUS:84925290985
SN - 0925-1022
VL - 75
SP - 127
EP - 144
JO - Designs, Codes, and Cryptography
JF - Designs, Codes, and Cryptography
IS - 1
ER -