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

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