MacWilliams' extension theorem for bi-invariant weights over finite principal ideal rings

Marcus Greferath, Thomas Honold, Cathy Mc Fadden, Jay A. Wood, Jens Zumbrägel

Research output: Contribution to journalArticleScientificpeer-review

7 Citations (Scopus)

Abstract

A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R -linear codes in RnRn extends to a monomial transformation of RnRn that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius.

This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property. Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal rings, which form a large subclass of the finite Frobenius rings. We do not assume commutativity of the rings in question.
Original languageUndefined/Unknown
Pages (from-to)177-193
Number of pages17
JournalJournal of Combinatorial Theory Series A
Volume125
DOIs
Publication statusPublished - 2014
MoE publication typeA1 Journal article-refereed

Keywords

  • Frobenius ring
  • principal ideal ring
  • linear code
  • extension theorem
  • Möbius function

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