Junru Zheng

On Boston bound for cyclic codes by DFT with unknown elements

The Boston bound is defined by the subset of defining set for cyclic codes. The authors proposed new simple derivation for the BCH bound, the HT bound and the shift bound, using the discrete Fourier transform (DFT) and the Blahut theorem. In this paper, we consider the Boston bound for cyclic codes by the DFT.The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. In this paper we consider cyclic codes defined by defining sequence and new simple derivation using the discrete Fourier transform with unknown elements and the Blahut theorem is shown. Moreover two examples of binary cyclic codes are given.

An algorithm for new lower bound of minimum distance by DFT for cyclic codes

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.

On relationship between proposed lower bound and shift bound for cyclic codes

For cyclic codes we proposed a new lower bound of minimum distance by discrete Fourier transform (DFT). In this paper, we will show relationships between the proposed bound and the shift bound by numerical experiments. Moreover, the computational complexity of proposed lower bound is compared with the shift bound.

A note a constant weight sequences over q-ary from cyclic difference sets

A method of construction for non- binary periodic sequences from the cyclic difference set is proposed. Two numerical examples are given, and their value distributions and some Hamming distances between self-cyclic shift and cross-cyclic shift of two sequences are evaluated.

A set of sequences over Z 5 with period 21 including almost highest linear complexities

A method of construction for non-binary periodic sequence with constant weight from cyclic difference set is proposed. Two numerical examples are given, and their distribution and some Hamming distance between self-cyclic shift and cross-cyclic shift of two sequences are evaluated. For a set of sequences over Z5 with period 21, we show that its elements have almost the highest linear complexities from second numerical example.

On relationship between proposed lower bound and well-known bounds for cyclic codes

There have been useful lower bounds for the minimum distance of a cyclic code, BCH bound, Hartmann-Tzeng bound, Roos bound and Shift bound. The authors proposed a lower bound of minimum distance by discrete Fourier transform (DFT) in 2010. The relationship between BCH bound, Hartmann-Tzeng bound and Roos bound, BCH bound, Hartmann-Tzeng bound and Shift bound are known, respectively. However, its unknown the relationship between Roos bound, Shift bound and Proposed bound. In this paper, we discuss this relation with the cyclic codes.

On some algorithms on the proposed lower bound of the designed minimum distance for cyclic codes

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.

Hamming distance correlation for q-ary constant weight codes

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 Z5 with 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 Z5 with 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.

On the consecutive sets of defining sequence for lower bounds on cyclic codes

For cyclic codes, the defining sequence is an important parameter, because the calculation of some well-known lower bounds are used it. In this paper, we consider the consecutive elements pattern of defining sequence and show its some properties through Roos bound.

On linear complexity of periodic sequences over extension fields from cyclic difference sets

The set of constant-weight sequences over GF(q) from the cyclic difference set generalized by the authors are considered. For the linear complexity(LC) of infinite sequences with their one period as an element in the set, we give a conjecture that LCs of all sequences except two in the set are the maximum as same as their period and LCs of remaining two sequences are the maximum value minus one. Five numerical examples over two prime fields and three non-prime(extension) fields are shown for evidences of main conjecture in this paper.