Finite-Ring Combinatorics and MacWilliams' Equivalence Theorem

M. Greferath, S. E. Schmidt

Research output: Contribution to journalArticleScientificpeer-review

77 Citations (Scopus)

Abstract

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.
Original languageUndefined/Unknown
Pages (from-to)17-28
Number of pages12
JournalJournal of Combinatorial Theory Series A
Volume92
Issue number1
DOIs
Publication statusPublished - 2000
MoE publication typeA1 Journal article-refereed

Keywords

  • codes over rings
  • MacWilliams' equivalence theorem
  • homogeneous weights
  • real functions on modules
  • Möbius inversion on posets
  • Frobenius rings

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