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On the Hamming distance of linear codes over a finite chain ring

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journal contribution
posted on 21.08.2006 by G.H. Norton, Ana Salagean
Let R be a finite chain ring (e.g. a Galois ring), K its residue field and C a linear code over R. We prove that d(C), the Hamming distance of C, is d((C : α)), where (C : α) is a submodule quotient, α is a certain element of R and — denotes the canonical projection to K. These two codes also have the same set of minimal codeword supports. We explicitly construct a generator matrix/polynomial of (C : α) from the generator matrix/polynomials of C. We show that in general d(C) ≤ d(C) with equality for free codes (i.e. for free R- submodules of Rn) and in particular for Hensel lifts of cyclic codes over K. Most of the codes over rings described in the literature fall into this class. We characterise MDS codes over R and prove several analogues of properties of MDS codes over finite fields. We compute the Hamming weight enumerator of a free MDS code over R.

History

School

  • Science

Department

  • Computer Science

Pages

290294 bytes

Citation

NORTON and SALAGEAN, 2000. On the Hamming distance of linear codes over a finite chain ring. IEEE transactions on information theory, 46 (3), pp. 1060-1067

Publisher

© IEEE Transactions on Information Theory Society

Publication date

2000

Notes

This article was published in the journal, IEEE transactions on information theory [© Transactions on Information Theory Society] and is also available at: http://ieeexplore.ieee.org/servlet/opac?punumber=18. [©2006 IEEE] Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

ISSN

0018-9448

Language

en

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