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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. Index Terms--Quasi-cyclic LDPC code, location-vector, dispersion, superposition, Berlekamp-Massey algorithm, Kotter-Vardy algorithm.

This paper presents five methods for constructing nonbinary LDPC codes based on finite geometries. These methods result in five classes of nonbinary LDPC codes, one class of cyclic LDPC codes, three classes of quasi-cyclic LDPC codes and one class of structured regular LDPC codes. Experimental results show that constructed codes in these classes decoded with iterative decoding based on belief propagation perform very well over the AWGN channel and they achieve significant coding gains over Reed-Solomon codes of the same lengths and rates with either algebraic hard-decision decoding or KotterVardy algebraic soft-decision decoding at the expense of a larger decoding computational complexity. Index Terms--Euclidean geometry, LDPC code, incidencevector, location-vector, Berlekamp-Massey algorithm, KotterVardy algebraic soft-decision algorithm

In the late 1950s and early 1960s, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for hard-decision algebraic decoding, such as Bose--Chaudhuri--Hocquenghem (BCH) and Reed--Solomon (RS) codes. This paper shows that finite fields can also be successfully used to construct algebraic low-density parity-check (LDPC) codes for iterative soft-decision decoding. Methods of construction are presented. LDPC codes constructed by these methods are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN), binary random, and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor, and rate of decoding convergence, collectively. Particularly, they have low error floors. Since the codes are QC, they can be encoded using simple shift registers with linear complexity. Index Terms--Burst, dispersion, erasure, error floor, low-density parity-check (LDPC) codes, masking, quasi-cyclic (QC) codes, zero span.

This paper is concerned with construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes for three different types of channels: the additive white Gaussian noise, the binary random erasure, and the binary burst erasure channels. Two algebraic methods for systematic construction of QC-LDPC codes are presented. Codes constructed perform well over all three types of channels. Index Terms--Erasure burst, erasure correction (EC), Euclidean geometry, iterative decoding, low-density parity-check (LDPC) codes, quasi-cyclic (QC) codes.

Quasi-cyclic (QC) low-density parity-check (LDPC) codes form an important subclass of LDPC codes. These codes have encoding advantage over other types of LDPC codes. This paper addresses the issue of efficient encoding of QC-LDPC codes. Two methods are presented to find the generator matrices of QC-LDPC codes in systematic-circulant (SC)form from their parity-check matrices, given in circulant form. Based on the SC form of the generator matrix of a QC-LDPC code, various types of encoding circuits using simple shift registers are devised. It is shown that the encoding complexity of a QC-LDPC code is linearly proportional to the number of parity bits of the code for serial encoding, and to the length of the code for high-speed parallel encoding. Index Terms--Array of circulants, quasi-cyclic (QC) low-density parity-check (LDPC) codes, systematic-circulant (SC) form.

This paper presents a superposition method for constructing low-density parity-check (LDPC) codes. Several classes of structured LDPC codes are constructed. Codes in these classes perform well with iterative decoding, and their Tanner graphs have girth at least six. Index Terms--Quasi-cyclic low-density parity-check (LDPC) codes, Reed-Solomon (RS) codes, superposition.

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 sumproduct 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.