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GeneralizedStability Condition forGeneralizedand Doubly-GeneralizedLDPC Codes

Enrico Paolini

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

Inthis paper, thestability condition forlow-density parity-check (LDPC)codes onthebinary erasure channel (BEC) isextended togeneralized LDPC (GLDPC)codesanddoubly- generalized LDPC (D-GLDPC) codes. Itisproved that, inboth cases, thestability condition onlyinvolves thecomponent codes withminimumdistance 2.Thestability condition forGLDPC codesisalways expressed asanupperboundtothedecoding threshold. Thisisnotpossible forD-GLDPCcodes, unless all thegeneralized variable nodeshaveminimumdistance atleast 3.Furthermore, acondition called derivative matching isdefined inthepaper. Thiscondition issufficient fora GLDPC orD- GLDPC codetoachieve thestability condition withequality. Ifthiscondition issatisfied, thethreshold ofD-GLDPCcodes (whose generalized variable nodeshaveallminimumdistance at least 3)andGLDPC codescanbeexpressed inclosed form.

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Inthis paper, thestability condition forlow-density parity-check (LDPC)codes onthebinary erasure channel (BEC) isextended togeneralized LDPC (GLDPC)codesanddoubly- generalized LDPC (D-GLDPC) codes. Itisproved that, inboth cases, thestability condition onlyinvolves thecomponent codes withminimumdistance 2.Thestability condition forGLDPC codesisalways expressed asanupperboundtothedecoding threshold. Thisisnotpossible forD-GLDPCcodes, unless all thegeneralized variable nodeshaveminimumdistance atleast 3.Furthermore, acondition called derivative matching isdefined inthepaper. Thiscondition issufficient fora GLDPC orD- GLDPC codetoachieve thestability condition withequality. Ifthiscondition issatisfied, thethreshold ofD-GLDPCcodes (whose generalized variable nodeshaveallminimumdistance at least 3)andGLDPC codescanbeexpressed inclosed form.
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 doubly-generalized 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 D- GLDPC 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.
The iterative decoding threshold of low-density parity-check (LDPC) codes over the binary erasure channel (BEC) fulfills an upper bound depending only on the variable and check nodes with minimum distance 2. This bound is a consequence of the stability condition, and is here referred to as stability bound. In this paper, a stability bound over the BEC is developed for doubly-generalized LDPC codes, where variable and check nodes can be generic linear block codes, assuming maximumaposteriorierasure correction at each node. It is proved that also in this generalized context the bound depends only on the variable and check component codes with minimum distance 2. A condition is also developed, namely, the derivative matching condition, under which the bound is achieved with equality. The stability bound leads to consider single parity-check codes used as variable nodes as an appealing option to overcome common problems created by generalized check nodes.
A generalized low-density parity check code (GLDPC) is a low-density parity check code in which the constraint nodes of the code graph are block codes, rather than single parity checks. In this paper, we study GLDPC codes which have BCH or Reed-Solomon codes as subcodes under bounded distance decoding (BDD). The performance of the proposed scheme is investigated in the limit case of an infinite length (cycle free) code used over a binary erasure channel (BEC) and the corresponding thresholds for iterative decoding are derived. The performance of the proposed scheme for finite code lengths over a BEC is investigated as well. Structures responsible for decoding failures are defined and a theoretical analysis over the ensemble of GLDPC codes which yields exact bit and block error rates of the ensemble average is derived. Unfortunately this study shows that GLDPC codes do not compare favorably with their LDPC counterpart over the BEC. Fortunately, it is also shown that under certain conditions, objects identified in the analysis of GLDPC codes over a BEC and the corresponding theoretical results remain useful to derive tight lower bounds on the performance of GLDPC codes over a binary symmetric channel (BSC). Simulation results show that the proposed method yields competitive performance with a good decoding complexity trade-off for the BSC.
In this paper, a method for the asymptotic analysis of generalized low-density parity-check (GLDPC) codes on the binary erasure channel (BEC) is proposed. The considered GLDPC codes have block linear codes as check nodes. Instead of considering specific check com-ponent codes, like Hamming or BCH codes, random codes are considered, and a technique is developed for obtaining the expected check EXIT function for the overall GLDPC code. Each check component code is supposed to belong to an expurgated ensemble. Some GLDPC thresholds obtained by this technique are com-pared with those of GLDPC codes, with the same dis-tribution and component codes lengths, using specific codes. Results obtained by combining our analysis with differential evolution tool are also presented. 1.
Protograph low-density-parity-check (LDPC) are considered to design near-capacity low-rate codes over the binary erasure channel (BEC) and binary additive white Gaussian noise (BIAWGN) channel. For protographs with degree-one variable nodes and doubly-generalized LDPC (DGLDPC) codes, conditions are derived to ensure equality of bit-error threshold and block-error threshold. Using this condition, low-rate codes with block-error threshold close to capacity are designed and shown to have better error rate performance than other existing codes.
Protograph doubly-generalized low-density parity-check (DGLDPC) codes, which allow for arbitrary component codes at the variable and check nodes of a protograph, are considered. Exact density evolution is derived over the binary erasure channel. Conditions on the protograph and component codes to ensure equality of block-error threshold and density evolution threshold for large-girth ensembles are established. Conditions for stability of density evolution are derived, and block-error threshold property is extended to binary-input symmetric channels. Optimized low-rate protographs for DGLDPC codes over the erasure channel are presented.