The minimum distance for linear codes is one of the important parameters. The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. It is necessary to construct the maximum number of the independent set for the calculation of the shift bound. However, its computational complexity is very large, because the construction of the independent sets is not unique. The authors proposed an algorithm for calculation of the independent set and new lower bound using the discrete Fourier transform in 2010. In this paper we give simple modification and new recurrent algorithms to improve the original algorithm.

We proposed a method for q-ary constant weight codes from the cyclic difference set by generalization of the method in binary case proposed by N. Li, X. Zeng and L. Hu in 2008. It was shown that two sets of constant weight codes over Z5with length 21 from the (21, 5, l)-planar (cyclic) difference set and constant weight codes over G F (8) with length 57 from the (57, 8, l)-planar (cyclic) difference set have almost highest linear complexities and good profiles of their linear complexities. Moreover we investigated the value distributions in all codewords with length 57 over G F (8) from the (57, 8, l)-planar difference set. It was pointed out that this set of periodic sequences also has good value distributions and almost highest linear complexities in similar to previous set of sequences over Z5with period 21. In this paper we calculate the Hamming distance between all distinct cyclic shift of themselves, called the auto-Hamming correlation and the Hamming distance between distinct codewords with all cyclic shift, called the cross-Hamming correlation. Consequently it is shown that all of the auto and cross-Hamming correlations are large against their code length for all codewords over G F (8) with length 57 from a (57, 8, l)-planar difference set.

For cyclic codes some well-known lower bounds and some decoding methods up to the half of the bounds are suggested. Particularly, the shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. However, the computational complexity of the shift bound is very large. In this paper we consider cyclic codes defined by their defining set, and a new method of the minimum distance using the discrete Fourier transform(DFT) is shown. Moreover some examples of binary cyclic codes are given.