FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
1997, VOLUME 3, NUMBER 1, PAGES 195-254
A. A. Nechaev
A. S. Kuzmin
V. T. Markov
Abstract
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The foundations of linear code theory over finite rings and modules
are developed. The main objects of investigation are: systematic code,
dual code, McWilliams identity, parity-check matrix an the Hamming
distance of a code. The properties of codes over modules and linear
spaces are compared, and the representations of linear codes by
polylinear recurrences are described, the latter being the most efficient
for systematic and Abelian group codes. The special role of
quasi-Frobenius modules in code theory is revealed. As corollaries
we obtain and generalize some known results. In particular, we build
cyclic Hamming and BCH codes over an arbitrary primary module.
All articles are published in Russian.
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