It is shown that Euclidean geometry LDPC codes in conjunction with their shortened codes obtained by puncturing their parity-check matrices are effective in correcting combinations of errors and erasures with a two-phase decoding scheme. This is due to the large row redundancies of the parity-check matrices of these codes which are given by the incidence matrices of Euclidean geometries.

Many known constructions of LDPC codes can be placed in a general framework using the notion of partial geometries. Based on this notion, the structure of such LDPC codes can be analyzed using a geometric approach that illuminates important properties of their parity-check matrices. In this approach, trapping sets are represented by subgeometries of the geometry used to construct the code. Based on the incidence relations between lines and points in this geometry, the structure of trapping sets is investigated. On the other hand, it is shown that removing a subgeometry corresponding to a trapping set gives a punctured matrix which can be used as a parity-check matrix of an LDPC code. This relates trapping sets, represented by subgeometries, and punctured matrices, represented by the residual geometries. The null spaces of these punctured matrices are LDPC codes which inherit many of the good structural properties of the original code. Hence, new LDPC codes, with various lengths and rates, can be obtained by puncturing an LDPC code constructed based on a partial geometry. Furthermore, these punctured matrices and codes can be used in a two-phase decoding scheme to correct combinations of errors and erasures.

The trapping set structure of LDPC codes constructed using finite geometries is analyzed. A trapping set is modeled as a sub-geometry of the geometry used to construct an LDPC code. The variable nodes of a trapping set are viewed as points of the geometry and the check nodes adjacent to the variable nodes are viewed as the lines passing through any of these points. Based on this geometrical representation of a trapping set, its configuration can be determined.

This paper presents algebraic methods for constructing high performance and efficiently encodable non-binary quasi-cyclic LDPC codes based on flats of finite Euclidean geometries and array masking. Codes constructed based on these methods perform very 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 either algebraic hard-decision Berlekamp-Massey algorithm or algebraic soft-decision Kotter-Vardy algorithm. Due to their quasi-cyclic structure, these non-binary LDPC codes on Euclidean geometries can be encoded using simple shift-registers with linear complexity. Structured non-binary LDPC codes have a great potential to replace Reed-Solomon codes for some applications in either communication or storage systems for combating mixed types of noise and interferences.

This paper presents new methods for constructing efficiently encodable quasi-cyclic LDPC codes based on special balanced incomplete block designs (BIBD's). Codes constructed perform well over both the AWGN and binary erasure channels with iterative decoding.

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 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 Kotter-Vardy algebraic soft-decision decoding at the expense of a larger decoding computational complexity.

This paper presents new methods for efficiently constructing encodable quasi-cyclic low-density parity-check (LDPC) codes based on special balanced incomplete block designs (BIBDs). Codes constructed perform well over both the additive white Gaussian noise (AWGN) and binary erasure channels with iterative decoding.

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.

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.

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

This paper presents three algebraic methods for constructing q-ary LDPC codes. The first method gives a class of dispersed Reed-Solomon codes as LDPC codes. The second method gives a class of q-ary quasi-cyclic LDPC codes. The third method gives two classes of q-ary finite geometry LDPC codes. Codes constructed by these methods perform very well with iterative decoding, even for short codes.

This paper presents two algebraic methods for constructing efficiently encodable quasi-cyclic (QC) LDPC codes that perform well on both the AWGN and binary erasure channels with iterative decoding in terms of bit-error performance, block error performance and error-floor, collectively. The constructions are based on the cyclic subgroups of the multiplicative groups of finite fields and affine mappings

This paper presents a method for constructing structured LDPC codes based on finite fields and affine mappings. The Tanner graphs of these codes have girth at least six. Experimental results show that codes constructed based on this method with iterative decoding perform well on both the AWGN and binary erasure channels. Furthermore, codes constructed based on prime fields are quasi-cyclic.