Projective systems and perfect codes with a poset metric. (English) Zbl 1047.51003
Author’s abstract: The notion of a projective system, defined as a set \(X\) of \(n\)-points in a projective space over a finite field, was introduced by Tsfasman and Vldut. By this notion, the weight distribution of a nondegenerate linear code can be computed by the configuration of \(X\) and hyperplanes in the dual projective space of \(C\).
In this paper, we construct a projective system \(X\) defined as a set of \(n\)-linear subspaces on the dual projective space of a poset code. This gives a natural one-to-one correspondence between the set of equivalence classes of nondegenerate poset projective systems and the set of nondegenerate poset codes. By this correspondence, the weight distribution of a nondegenerate poset code can be also computed by the configuration of \(X\) and hyperplanes in the dual projective space of \(C\). Finally, we provide an algorithm for finding perfect poset codes.
In this paper, we construct a projective system \(X\) defined as a set of \(n\)-linear subspaces on the dual projective space of a poset code. This gives a natural one-to-one correspondence between the set of equivalence classes of nondegenerate poset projective systems and the set of nondegenerate poset codes. By this correspondence, the weight distribution of a nondegenerate poset code can be also computed by the configuration of \(X\) and hyperplanes in the dual projective space of \(C\). Finally, we provide an algorithm for finding perfect poset codes.
Reviewer: Giorgio Faina (Perugia)
References:
| [1] | Brualdi, R.; Graves, J. S.; Lawrence, M., Codes with a poset metric, Discrete Math., 147, 57-72 (1995) · Zbl 0854.94019 |
| [2] | Hirschfeld, J. W.P.; Tsfasman, M. A.; Vlǎdut, S. G., The weight hierarchy of higher-dimensional Hermitian codes, IEEE Trans. Inform. Theory, 40, 275-278 (1994) · Zbl 0802.94015 |
| [3] | Tsfasman, M. A.; Vlǎdut, S. G., Algebraic Geometry Codes (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0727.94007 |
| [4] | Tsfasman, M. A.; Vlǎdut, S. G., Geometric approach to higher weights, IEEE Trans. Inform. Theory, 41, 1564-1588 (1995) · Zbl 0853.94023 |
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