Finite-Ring Combinatorics and MacWilliams' Equivalence Theorem
| Autor: | S.E. Schmidt, Marcus Greferath |
|---|---|
| Jazyk: | angličtina |
| Předmět: |
MacWilliams' equivalence theorem
Finite ring Monomial homogeneous weights Theoretical Computer Science Combinatorics Computational Theory and Mathematics Möbius inversion on posets Homogeneous real functions on modules codes over rings Discrete Mathematics and Combinatorics Equivalence (formal languages) Partially ordered set Mobius inversion Hamming code Frobenius rings Mathematics |
| Zdroj: | Journal of Combinatorial Theory, Series A. (1):17-28 |
| ISSN: | 0097-3165 |
| DOI: | 10.1006/jcta.1999.3033 |
| Popis: | F. J. MacWilliams proved that Hamming isometries between linear codes extend to monomial transformations. This theorem has recently been genera- lized by J. Wood who proved it for Frobenius rings using character theoretic methods. The present paper provides a combinatorial approach: First we extend I. Constantinescu's concept of homogeneous weights on arbitrary finite rings and prove MacWilliams' equivalence theorem to hold with respect to these weights for all finite Frobenius rings. As a central tool we then establish a general inversion principle for real functions on finite modules that involves Möbius inversion on partially ordered sets. An application of the latter yields the aforementioned result of Wood. |
| Databáze: | OpenAIRE |
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