Studying the locator polynomials of minimum weight codewords of BCH codes

We consider only primitive binary cyclic codes of length n = 2 m ? 1. A BCH-code with designed distance is denoted B(n;). A BCH-code is always a narrow-sense BCH-code. A codeword is identiied with its locator polynomial, whose coeecients are the symmetric functions of the locators. The deenition 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 with the Newton's identities. We rst present an algebraic point of view in order to prove or innrm the existence of words of a given weight in a code. The main tool is a symbolic computation software in exploring the Newton's identities. Our principal result is the true minimum distance of some BCH-codes of length 255 and 511, which were not known. In a second part, we study the codes B(n; 2 h ? 1), h 2 3; m ? 2]. We prove 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. We give some Corollaries of this result.