The minimum distance of some binary codes via the Newton's identities

In this paper, we give a natural way of deciding whether a given cyclic code contains a word of given weight. The method is based on the manipulation of the locators and of the locator polynomial of a codeword x. Because of the dimensions of the problem, we need to use a symbolic computation software, like Maple or Scratchpad II. The method can be ineeective when the length is too large. The paper contains two parts : In the rst part we will present the main deenitions and properties we need. In the second part, we will explain how to use these properties, and, as illustration, we will prove the three following facts : The dual of the BCH code of length 63 and designed distance 9 has true minimum distance 14 (which was already known). The BCH code of length 1023 and designed distance 253 has minimum distance 253. The cyclic codes of length 2 11 ?1; 2 13 ?1; 2 17 ?1, with generator polynomial m 1 (x) and m 7 (x) have minimum distance 4 (see 5]). 1 Notation and properties. 1.1 Cyclic codes. We use the following deenition for cyclic codes : Deenition 1 A primitive cyclic code C of length n = 2 m ? 1 over GF(2) is an ideal of the polynomial algebra GF(2))x]=(x n ?1). The roots of its generator polynomial are called the zeros of C. The code C is characterized by its deening set I(C) : I(C) = fi 2 0::n ? 1] j i is a zero of Cg (1) where is a n th root of unity in GF(2 m).