MacWilliams’ extension theorem for infinite rings
نویسندگان
چکیده
منابع مشابه
MacWilliams' extension theorem for infinite rings
Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle em...
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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 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 E...
متن کاملMacWilliams Extension Theorems and the Local-Global Property for Codes over Rings
The MacWilliams extension theorem is investigated for various weight functions over finite Frobenius rings. The problem is reformulated in terms of a local-global property for subgroups of the general linear group. Among other things, it is shown that the extension theorem holds true for poset weights if and only if the underlying poset is hierarchical. Specifically, the Rosenbloom-Tsfasman wei...
متن کاملSemigroup Rings and the Extension Theorem for Linear Codes
An extension theorem for general weight functions is proved over nite commutative local principal ideal rings. The structure of the complex semigroup ring associated to the multiplicative semigroup of the ring plays a prominent role in the proof. 1. Background In her doctoral dissertation, MacWilliams [8], [9] proved an equivalence theorem: two linear codes C1; C2 F n de ned over a nite eld F a...
متن کاملMacWilliams Extension Theorem for MDS codes over a vector space alphabet
The MacWilliams Extension Theorem states that each linear isometry of a linear code extends to a monomial map. Unlike the linear codes, in general, additive codes do not have the extension property. In this paper, an analogue of the extension theorem for additive codes in the case of additive MDS codes is proved. More precisely, it is shown that for almost all additive MDS codes their additive ...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2018
ISSN: 0002-9939,1088-6826
DOI: 10.1090/proc/14343