Generalized Stability Condition for Generalized and Doubly-Generalized LDPC Codes

In this paper, the stability condition for low-density parity-check (LDPC) codes on the binary erasure channel (BEC) is extended to generalized LDPC (GLDPC) codes and doublygeneralized LDPC (D-GLDPC) codes. It is proved that, in both cases, the stability condition only involves the component codes with minimum distance 2. The stability condition for GLDPC codes is always expressed as an upper bound to the decoding threshold. This is not possible for D-GLDPC codes, unless all the generalized variable nodes have minimum distance at least 3. Furthermore, a condition called derivative matching is defined in the paper. This condition is sufficient for a GLDPC or DGLDPC code to achieve the stability condition with equality. If this condition is satisfied, the threshold of D-GLDPC codes (whose generalized variable nodes have all minimum distance at least 3) and GLDPC codes can be expressed in closed form.Comment: 5 pages, 2 figures, to appear in Proc. of IEEE ISIT 200

On a Class of Doubly-Generalized LDPC Codes with Single Parity-Check Variable Nodes

A class of doubly-generalized low-density parity-check (D-GLDPC) codes, where single parity-check (SPC) codes are used as variable nodes (VNs), is investigated. An expression for the growth rate of the weight distribution of any D-GLDPC ensemble with a uniform check node (CN) set is presented at first, together with an analytical technique for its efficient evaluation. These tools are then used for detailed analysis of a case study, namely, a rate-1/2 D-GLDPC ensemble where all the CNs are (7,4) Hamming codes and all the VNs are length-7 SPC codes. It is illustrated how the VN representations can heavily affect the code properties and how different VN representations can be combined within the same graph to enhance some of the code parameters. The analysis is conducted over the binary erasure channel. Interesting features of the new codes include the capability of achieving a good compromise between waterfall and error floor performance while preserving graphical regularity, and values of threshold outperforming LDPC counterparts.Comment: 2009 IEEE Int. Symp. on Information Theory. 5 pages, 3 figure

On design of doubly-generalized LDPC codes based on multi-type information functions

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Information-bit error rate and false positives in an MDS code

In this paper, a refinement of the weight distribution in an MDS code is computed. Concretely, the number of codewords with a fixed amount of nonzero bits in both information and redundancy parts is obtained. This refinement improves the theoretical approximation of the information-bit and -symbol error rate, in terms of the channel bit-error rate, in a block transmission through a discrete memoryless channel. Since a bounded distance reproducing encoder is assumed, the computation of the here-called false positive (a decoding failure with no information-symbol error) is provided. As a consequence, a new performance analysis of an MDS code is proposed

Quasi-cyclic LDPC codes of column-weight two using a search algorithm

Copyright © 2007 G. Malema and M. Liebelt. This is an Open Access article distributed under the Creative Commons Attributions License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.This article introduces a search algorithm for constructing quasi-cyclic LDPC codes of column-weight two. To obtain a submatrix structure, rows are divided into groups of equal sizes. Rows in a group are connected in their numerical order to obtain a cyclic structure. Two rows forming a column must be at a specified distance from each other to obtain a given girth. The search for rows satisfying the distance is done sequentially or randomly. Using the proposed algorithm regular and irregular column-weight-two codes are obtained over a wide range of girths, rates, and lengths. The algorithm, which has a complexity linear with respect to the number of rows, provides an easy and fast way to construct quasi-cyclic LDPC codes. Constructed codes show good bit-error rate performance with randomly shifted codes performing better than sequentially shifted ones.Gabofetswe Malema and Michael Liebel

Pseudo-binomial degree distributions from derivative matching

In this paper, a method to design check-concentrated LDPC degree distributions for the erasure channel is proposed. This method is obtained taking a derivative matching approach. It consists of matching the first and high-order derivatives of the variable node decoder EXIT function and inverse check node decoder EXIT function in order to reduce the gap between the two curves in the EXIT chart. A sufficient condition for a check-concentrated distribution to achieve derivative matching up to some order is first developed, and then a design algorithm is proposed exploiting this sufficient condition. A comparison with other deterministic design techniques is provided, revealing the potentialities of the proposed algorithm

On the Design of Irregular GLDPC Codes with Low Error Floor over the BEC

The design of GLDPC codes for the binary erasure channel with low error floor under iterative decoding is investigated. Both bounded distance and maximum a posteriori decoding at the check nodes are considered. For both check node decoding algorithms a key para- meter is identified, discriminating between an exponen- tially small and exponentially large expected number of small size stopping sets. A code design technique is proposed based on this theoretical investigation

On the growth rate of irregular GLDPC codes weight distribution

In this paper the exponential growth rate of irregular generalized low-density parity-check (GLDPC) codes weight distribution is considered. Specifically, the Taylor series of the growth rate is expanded to the first order with the purpose of studying its behavior in correspondence with the small weight codewords. It is proved that the linear term of the Taylor series, and then the expected number of small linear-sized weight codewords of a randomly chosen GLDPC code in the irregular ensemble, is dominated by the degree-2 variable nodes and by the check nodes with minimum distance 2. A parameter is introduced, only depending on such variable and check nodes, discriminating between an exponentially small and exponentially large expected number of small weight codewords