MacWilliams' Extension Theorem for bi-invariant weights over finite principal ideal rings

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:125

Article

Greferath, Marcus¹  Honold, Thomas²  Mc Fadden, Cathy¹  Wood, Jay A.³  Zumbraegel, Jens¹ 

[1] Univ Coll Dublin, Claude Shannon Inst, Sch Math Sci, Dublin 4, Ireland

[2] Zhejiang Univ, Inst Informat & Commun Engn, Dept Informat Sci & Elect Engn, Hangzhou 310027, Zhejiang, Peoples R China

[3] Western Michigan Univ, Dept Math, Kalamazoo, MI 49008 USA

关键词: Frobenius ring;    Principal ideal ring;    Linear code;    Extension Theorem;    Mobius function;   

DOI  :  10.1016/j.jcta.2014.03.005

来源: Elsevier

【 摘 要 】

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 R-n extends to a monomial transformation of R-n 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. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free