Idempotent and Bch Bound

Using the characterization of the idempotents of a narrow-sense primitive binary BCH code, we are able to give classes of such codes whose minimum distance reaches the BCH bound. Let be a primitive element of GF(2 m). We denote R n = GF(2))x]=(x n ? 1), n = 2 m ? 1. 1 Binary BCH codes and locator polynomials 1.1 Primitive binary BCH codes For our purpose we only need to deene a particular class of BCH codes: the primitive narrow-sense BCH codes. Deenition 1 The primitive narrow-sense binary BCH code of length n and designed distance , denoted B(n;) if it exists, is the set S of polynomials of R n such that: 1. For all c(x) in S and for all i, 1 i < , c(i) = 0. 2. There exists g(x) in S such that g() 6 = 0. Remark 1 1. A code may verify the rst condition of Deenition 1 but not the second. This happens if there is a conjugate j of in GF(2 m) such that 1 j <. In this case the code is equal to B(n; 0) for some 0 > and B(n;) does not exist. 2. A necessary and suucient condition for B(n;) to exist is that is the smallest element of its cyclotomic class modulo n. In particular must be odd. 3. The deenition of B(n;) depends on the choice of a primitive element in GF(2 m), although it leads to an equivalent code. To get rid this problem, we consider that, once for all, we have chosen a primitive element for each extension of GF(2). Thus speaking of THE code B(2 s ? 1;) for all s and any suitable is sound.