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Studying the Locator Polynomials of Minimum Weight Codewords of BCH Codes

Abstract : Only primitive binary cyclic codes of length n = 2^m - 1 are considered. A BCH-code with designed distance delta is denoted B (n,delta ). A BCH-code is always a narrow-sense BCH-code. A codeword is identified with its locator polynomial, whose coefficients are the symmetric functions of the locators. The definition of the code by its zeros-set involves some properties for the power sums of the locators. Moreover, the symmetric functions and the power sums of the locators are related to Newton's identities. First presented is an algebraic point of view in order to prove or disprove the existence of words of a given weight in a code. The main tool is symbolic computation software to explore Newton's identities. The principal result is the true minimum distance of some BCH-codes of length 255 and 511, which were not known. In a second part, the minimum weight codewords of the codes B(n,2^(m-2) - 1) are studied. It is proven that the set of the minimum weight codewords of the BCH-code B(n, 2^(m-2)-1) equals the set of the minimum weight codewords of the punctured Reed-Muller code of length n and order 2, for any m. Several corollaries of this result are given.
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Daniel Augot, Pascale Charpin, Nicolas Sendrier. Studying the Locator Polynomials of Minimum Weight Codewords of BCH Codes. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 1992, 38 (3), pp.960-973. ⟨10.1109/18.135638⟩. ⟨hal-00723733⟩

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