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

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 Rn extends to a monomial transformation of Rn that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series A 2014-07, Vol.125, p.177-193
Main Authors: Greferath, Marcus, Honold, Thomas, Mc Fadden, Cathy, Wood, Jay A., Zumbrägel, Jens
Format: Article
Language:English
Subjects:
Extension Theorem
Frobenius ring
Linear code
Möbius function
Principal ideal ring
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Summary: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 Rn extends to a monomial transformation of Rn 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.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2014.03.005