Lecture Notes on Linear Codes Defined over Finite Modules: the Extension Theorem and the Macwilliams Identities — for Use of Cimpa-unesco-tübitak Summer
نویسنده
چکیده
These lecture notes discuss the extension problem and the MacWilliams identities for linear codes defined over finite modules.
منابع مشابه
Foundations of Linear Codes Defined over Finite Modules: the Extension Theorem and the Macwilliams Identities — Based on Lectures for the Cimpa-unesco-tübi̇tak Summer School
This paper discusses the foundations of the theory of linear codes defined over finite modules. Two topics are examined in depth: the extension theorem and the MacWilliams identities. Both of these topics were studied originally by MacWilliams in the context of linear codes defined over finite fields.
متن کاملDuality for Modules over Finite Rings and Applications to Coding Theory
This paper sets a foundation for the study of linear codes over nite rings. The nite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from nite elds to nite Frobenius rings. It is over Frobenius rings that certain key identi cations can be made bet...
متن کاملLinear codes over Z4+uZ4: MacWilliams identities, projections, and formally self-dual codes
Linear codes are considered over the ring Z 4 + uZ 4 , a non-chain extension of Z 4. Lee weights, Gray maps for these codes are defined and MacWilliams identities for the complete, symmetrized and Lee weight enumer-ators are proved. Two projections from Z 4 + uZ 4 to the rings Z 4 and F 2 + uF 2 are considered and self-dual codes over Z 4 +uZ 4 are studied in connection with these projections. ...
متن کاملLatroids and Their Representation by Codes over Modules
It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields. In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field. There are intimately related operations of deletion, contraction, minors and duality on both the code and the matroid. The weight enumerator of the code i...
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 کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008