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 presents algebraic methods for constructing high performance quasi-cyclic LDPC codes over non-binary fields. Experimental results show that codes constructed based on these methods perform well over the AWGN channel with iterative decoding using a fast Fourier transform based sum-product algorithm. They achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with the hard-decision Berlekamp-Massey algorithm, the algebraic soft-decision Kotter-Vardy algorithm, and the Jiang-Narayananpsilas adaptive belief propagation algorithm. Due to their quasi-cyclic structure, these LDPC codes can be efficiently encoded using simple shift-registers with linear complexity. They have a great potential to replace Reed-Solomon codes for some applications in communication or storage systems for combating mixed types of noise and interferences.

A vector quantization (VQ) scheme with finite memory called feature map finite-state vector quantization (FMFSVQ) is presented. The FMFSVQ takes advantage of good topological ordering so that the design of state codebooks is simplified. Our FMFSVQ also has no duplication of state codebooks, no synchronization required between encoder and decoder, and a very simple decoder. An adaptive FMFSVQ scheme is also proposed. Experimental results are presented for different super codebook sizes and different state codebook sizes.