A constructive method to find the cycles of an arbitrary cyclic (n,k)-code over IFq relevant to weight distribution and cross correlation

In order to determine the weights of an arbitrary cyclic (n,k)-code over IF_q, q=p^e, gcd(q,n)=1, p prime element of IN, it is necessary to know the cycles of that code. The codewords of a cycle are produced from each other by cyclic shifting. A representative codeword of a cycle is called a cycle-leader. If the code is irreducible, i.e. its parity check-polynomial (Pcp) is irreducible, and the degree k of the Pcp fulfils k=ord_n(q), the code has N=(q^k-1)/n cycles. For this case, a method is developed for finding a suitable irreducible (q^k-1,k)-cyclic code (“frame-code”), that contains, in a concatenated manner, the cycle-leaders of the cyclic code. For the case of a reducible code, an iterative method is explained in order to compute all cycle-leaders of the code. Besides determination of the weight distribution of an arbitrary cyclic code, this method also simplifies examining the cross-correlation properties of roots-of-unity-sequences derived from cyclic codes, since the cycle-leaders are necessary and sufficient for the values of the cross correlation function