An error-burst correcting algorithm is developed based on a circulant parity-check matrix of a cyclic code. The proposed algorithm is more efficient than error trapping if the code rate is less than about 2/3. It is shown that for any (n, k) cyclic code, there is an n × n circulant parity-check matrix such that the algorithm, applied to this matrix, corrects error bursts of lengths up to the error-burst correction limit of the cyclic code. This same matrix can be used to efficiently correct erasure bursts of lengths up to n - k. The error-burst correction capabilities of a class of cyclic low-density parity-check (LDPC) codes constructed from finite geometries are also considered.

A unified approach for constructing binary and nonbinary quasi-cyclic LDPC codes under a single framework is presented. Six classes of binary and nonbinary quasi-cyclic LDPC codes are constructed based on primitive elements, additive subgroups, and cyclic subgroups of finite fields. Numerical results show that the codes constructed perform well over the AWGN channel with iterative decoding.

This paper presents two algebraic methods for constructing high performance and efficiently encodable nonbinary quasi-cyclic LDPC codes based on arrays of special circulant permutation matrices and multi-fold array dispersions. Codes constructed based on these methods perform well over the AWGN and other types of channels with iterative decoding based on belief-propagation. Experimental results show that over the AWGN channel, these non-binary quasi-cyclic LDPC codes significantly outperform Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard-decision Berlekamp-Massey algorithm or algebraic soft-decision Kötter- Vardy algorithm. Also presented in this paper is a class of asymptotically optimal LDPC codes for correcting bursts of erasures. Codes constructed also perform well over flat fading channels. Non-binary quasi-cyclic LDPC codes have a great potential to replace Reed-Solomon codes in some applications in communication environments and storage systems for combating mixed types of noises and interferences.

This paper is concerned with construction of efficiently encodable nonbinary quasi-cyclic LDPC codes based on finite fields. Four classes of nonbinary quasi-cyclic LDPC codes are constructed. Experimental results show that codes constructed perform well with iterative decoding using a fast Fourier transform based q-ary sum-product algorithm and they achieve significant coding gains over Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard- decision Berlekamp-Massey algorithm or algebraic soft-decision Kotter-Vardy algorithm.

This paper investigates cyclic codes for correcting bursts of errors from a new point of view. A simple iterative algorithm for correcting bursts of errors is developed. This algorithm is optimal in the sense that it corrects burst of errors of lengths up to the burst-error-correction limit of a cyclic code. Also included in the paper is an iterative process for correcting bursts of erasures.

This paper presents a general and three specific algebraic methods for constructing efficiently encodable non-binary quasi-cyclic LDPC codes. Three classes of quasi-cyclic LDPC codes over nonbinary finite fields are constructed. codes constructed perform very well over the AWGN channel with iterative decoding and achieve large coding gains over the Reed-Solomon codes of the same parameters. Nonbinary LDPC codes may be used to replace Reed-Solomon codes in some communication environments or storage systems for combating mixed types of noises and interferences.

LDPC codes have excellent performance when iteratively decoded based on sparse parity-check matrices over both the AWGN channel and the erasure channel. In this paper, we propose a simple burst-correction decoding scheme for cyclic LDPC codes based on the same sparse circulant parity-check matrices used to perform iterative decoding. Although the proposed scheme may not achieve the full burst-correcting capability of the codes, it is rather simple and fast. We also show that the full burst-correcting capabilities of some codes constructed from Euclidean and projective geometries are very close to the Reiger upper bound

This paper presents three algebraic methods for constructing nonbinary quasi-cyclic (QC)-LDPC codes. Three classes of efficiently encodable QC-LDPC codes over nonbinary finite fields are constructed. Experimental results show that constructed codes decoded with iterative decoding perform well over the AWGN channel and they achieve significant coding gains over Reed-Solomon (RS) codes of the same lengths and rates decoded with algebraic decoding