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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Communicated by C. Carlet.
This work was supported by The National Science Foundation of China ( No. 11171366). The material of this paper was presented in part at the IEEE International Symposium on Information Theory, MIT, Cambridge, Boston, USA, July 2012.
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Liu, Z., Wu, XW. On relative constant-weight codes. Des. Codes Cryptogr. 75, 127–144 (2015). https://doi.org/10.1007/s10623-013-9896-2
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DOI: https://doi.org/10.1007/s10623-013-9896-2
Keywords
- Relative two-weight code
- Relative three-weight code
- Minimal codeword
- Finite projective geometry
- Wire-tap channel of type II
- Secret sharing