Simple proofs for duality of generalized minimum poset weights and weight distributions of (near-) MDS poset codes
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In 1991, Wei introduced generalized minimum Hamming weights for linear codes and showed their monotonicity and duality. Re-cently, several authors extended these results to the case of generalized minimum poset weights by using different methods. Here, we would like to prove the duality by using matroid theory. This gives yet another and very simple proof of it. In particular, our argument will make it clear that the duality follows from the well-known relation between the rank function and the corank function of a matroid. In addition, we derive the weight distributions of linear MDS and Near-MDS poset codes in the same spirit.
Abstract
In 1991, Wei introduced generalized minimum Hamming weights for linear codes and showed their monotonicity and duality. Re-cently, several authors extended these results to the case of generalized minimum poset weights by using different methods. Here, we would like to prove the duality by using matroid theory. This gives yet another and very simple proof of it. In particular, our argument will make it clear that the duality follows from the well-known relation between the rank function and the corank function of a matroid. In addition, we derive the weight distributions of linear MDS and Near-MDS poset codes in the same spirit.
참고문헌 (13)
- A. Barg, "Near-MDS poset codes and distributions, in: Error- Correcting Codes, Finite Geometries, and Cryptography", AMS Series: Contemporary Mathematics, 523, pp.135~147 (2010)
- R. A. Brualdi, "Codes with a poset metric", Discrete Math, 147, pp.57~72 (1995)
- S. Choi, "A new proof of MacWilliams identity for a hierarchical poset code", ~
- S. Dodunekov, "Near-MDS codes", J. Geom, 54, pp.30~43 (1995)
- S. Dodunekov, "Codes and projective multisets", Electron. J. Combin, 5(R37) (1998)
- T. Helleseth, "The weight distribution of irreducible cyclic codes with block lengths n1((ql ¡ 1)=N)", Discr. Math., 18, pp.179~211 (1977)
- 김현광, "Maximum distance separable poset codes", DESIGNS CODES AND CRYPTOGRAPHY, 48(3), pp.247~261, SPRINGER
- A. de Oliveira Moura, "Duality for poset codes", IEEE Trans. Inform. Theory, 56, pp.3180~3186 (2010)
- L. H. Ozarow, "Wire-tap-channel II", AT&T Bell Labs Tech. J, 63, pp.2135~2157 (1984)
- H. G. Schaathun, "Duality and support weights distributions", IEEE Trans. Inf. Theory, 50, pp.862~867 (2004)
- M. A. Tsfasman, "Geometric approach to higher weights", IEEE Trans. Inform. Theory, 41, pp.1564~1588 (1995)
- V. K. Wei, "Generalized Hamming weights for linear codes", IEEE Trans. Inform. Theory, 37, pp.1412~1418 (1991)
- D. J. A. Welsh, "Matroid Theory" (1976), Academic Press