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Description of minimum weight codewords of cyclic codes by algebraic systems

Daniel Augot 1 
1 CODES - Coding and cryptography
Inria Paris-Rocquencourt
Abstract : We consider cyclic codes of length n over ‫ކGF(q), n being prime to q. Fo such a cyclic code C, we describe a system of algebraic equations, denoted by SC(w), where w is a positive integer. The system is constructed from Newton's identities, which are satisfied by the elementary symmetric functions and the (generalized) power sum symmetric functions of the locators of codewords of weight w. The main result is that, in a certain sense, the algebraic solutions of SC(w) are in one-to-one correspondence with all the codewords of C having weight lower than w. In the particular case where w is the minimum distance of C, all minimum weight codewords are described by SC(w). Because the system SC (w) is very large, with many indeterminates, no great insight can be directly obtained, and specific tools are required in order to manipulate the algebraic systems. For this purpose, the theory of Groebner bases can be used. A Groebner basis of SC (w) gives information about the minimum weight codewords.
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Daniel Augot. Description of minimum weight codewords of cyclic codes by algebraic systems. Finite Fields and Their Applications, Elsevier, 1996, pp.138-152. ⟨10.1006/ffta.1996.0009⟩. ⟨hal-00723500⟩

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