The Weight Distributions Of Cyclic Codes

In coding theory, it is important to determine the weight distributions of codes. It is difficult to determine the weight distributions of cyclic codes. This paper mainly discuss the weight distributions of cyclic codes in special cases.First, let p be a prime, q=p8for a positive integer s, and γ=qm for a positive integer m. Let N>1be an integer dividing γ-1, and put n=(γ-1)/N. Let a be a primitive element of GF(γ) and let0=aN. We determine the weight distributions of irreducible cyclic code in some cases, where Trr/q is the trace function from GF(γ) to GF(q).Second, let p be a prime, q=ps, r=qm, where s and m are positive integers. Let u and v be two elements which are not conjugates of each other in GF(γ)*, and mu(x) and mv(x) be the minimal polynomials of u and v over GF(q) respectively. Define C(u, v, q, m): where a E GF(qhl), b E GF(qh2), h1and h2are the degree of mu(x) and mv(x) respec-tively, T1is the trace function from GF(γ) to GF(qhl), T2is the trace function from GF(γ) to GF(qh2). We calculate the weight distributions of C(u, v, q, m) in some special cases.