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Birkhäuser Basel eBooks

Combinatorially Designed LDPC Codes Using Zech Logarithms and Congruential Sequences

Jing Li

Published Jan 1, 2004• 0 citations
Publisher

Abstract

We investigate a systematic construction of regular low density parity check (LDPC) codes based on (γργ-1, ργ, ρ, γ{0,1}) combinatorial designs. The proposed (γ,ρ) regular LDPC ensemble has rate $${\left( {1 - \tfrac{1}{\rho }} \right)^\gamma }$$ girth ≥2γ+1, and exists for all γ≥2. The codes are a subset of Gallager’s random ensemble, but contains a good combination of structure and (pseudo)randomness. In particular, the simple case of γ=2 results in a class of codes that are high-rate, systematic, quasi-cyclic, linear-time encodable and decodable, and free of length-4 and length-6 cycles. Analysis on distance spectrum shows that they are better than the Gallager ensemble of the same parameters. Simulation of the proposed codes on intersymbol interference channels show that they perform comparably to random LDPC codes. Unlike random codes, the proposed structured LDPC codes can lend themselves to a low-complexity implementation for high-speed applications.

DOI
10.1007/978-3-0348-7865-4_12

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We investigate a systematic construction of regular low density parity check (LDPC) codes based on (γργ-1, ργ, ρ, γ{0,1}) combinatorial designs. The proposed (γ,ρ) regular LDPC ensemble has rate (1−1ρ)γ{\left( {1 - \tfrac{1}{\rho }} \right)^\gamma }(1−ρ1​)γ girth ≥2γ+1, and exists for all γ≥2. The codes are a subset of Gallager’s random ensemble, but contains a good combination of structure and (pseudo)randomness. In particular, the simple case of γ=2 results in a class of codes that are high-rate, systematic, quasi-cyclic, linear-time encodable and decodable, and free of length-4 and length-6 cycles. Analysis on distance spectrum shows that they are better than the Gallager ensemble of the same parameters. Simulation of the proposed codes on intersymbol interference channels show that they perform comparably to random LDPC codes. Unlike random codes, the proposed structured LDPC codes can lend themselves to a low-complexity implementation for high-speed applications.
Abstract Recent breakthroughs in forward error correction, in the form of low-density parity-check(LDPC) and turbo codes, have seen the promise of near Shannon limit information trans-mission realised. The deterministic construction of good codes, for LDPC codes in par-ticular, is currently the subject of intense interest in the research and development com-munity. This thesis adds to this field, developing methods and supporting theory for thedesign of algebraic LDPC codes using incidence structures from combinatorial designs.The benefits of this approach are guaranteed code properties, parameter flexibility, easeof implementation and reduced encoding complexity.In the first part of this thesis Steiner 2-designs and partial geometries are used toprovide systematic constructions of regular LDPC codes with Tanner graphs free of 4-cyclesandwithdeterministiccodeproperties. Drawingextensivelyfromexistingincidencestructures we present new families of algebraic LDPC codes with benefits over traditionalrandomly constructed regular codes, particularly where small-to-medium length codes areconcerned. We focus in particular on exploring the effect of different design propertieson the code parameters by deriving the rate, minimum distance, girth, and stopping setdistribution of LDPC codes from designs.In the second part of this thesis resolvable designs are employed to create regularLDPC codes free of 4-cycles with the parameter flexibility and decoding performance ofrandomly constructed regular LDPC codes. Employing resolvability greatly extends theclass of LDPC codes that can be algebraically constructed. These codes can offer sig-nificant implementation advantages due to the deterministic structure of their resolutionclasses and in some cases their simple encoding.A shortcoming of LDPC codes is their potentially high encoding complexity, whichis generally quadratic in the code length. Finding computationally efficient encodersis therefore critical for LDPC codes to be considered as serious contenders for futuregenerations of forward error correction devices. As a result the purpose of the last part ofthis thesis is to design codes which have the graph-based properties necessary to performwell with sum-product decoding but which are also cyclic or quasi-cyclic and so can beencoded simply. The main focus is regular codes but we also extend our constructionof quasi-cyclic codes to produce irregular LDPC codes with simple linear-time encodingcircuits.
Gallager introduced low-density parity-check (LDPC) codes in 1962, presenting a construction method to randomly allocate bits in the parity-check matrix subject to certain structural constraints. Since then improvements have been made to Gallager's construction method and some analytic constructions for LDPC codes have been presented. However analytically constructed LDPC codes comprise only a very small subset of possible codes and as a result LDPC codes are still, for the most part, constructed randomly. This paper extends the class of LDPC codes that can be systematically generated by presenting a construction method for regular LDPC codes based on combinatorial designs known as Kirkman triple systems. That is, we construct (3, /spl rho/)-regular codes whose Tanner (1981) graph is free of 4-cycles for any integer /spl rho/.
This paper introduces several new combinatorial constructions of low-density parity-check (LDPC) codes, in contrast to the prevalent practice of using long, random-like codes. The proposed codes are well structured, and unlike random codes can lend themselves to a very low-complexity implementation. Constructions of regular Gallager codes based on cyclic difference families, cycle-invariant difference sets, and affine 1-configurations are introduced. Several constructions of difference families used for code design are presented, as well as bounds on the minimal distance of the codes based on the concept of a generalized Pasch configuration.
Analytically constructed LDPC codes comprise only a very small subset of possible codes and as a result LDPC codes are still, for the most part, constructed randomly. This paper extends the class of LDPC codes that can be systematically generated by presenting a construction method for regular LDPC codes based on combinatorial designs known as Kirkman triple systems. We construct (3, /spl rho/)-regular codes whose Tanner graph is free of 4-cycles for any integer /spl rho/, and examine girth and minimum distance properties of several classes of LDPC codes obtained from combinatorial designs.
This paper,based on algebra and graph,proposes a method for constructing structured low-density parity-check(LDPC) codes.In this method,the authors design a connected graph with three kinds of special path to ensure that the Tanner graph of the parity check matrix mapped from the connected graph is without short cycles.The construction method results in two classes of(3,k)-regular quasi-cyclic LDPC codes with girth 8 and 12 respectively.Furthermore,by extending this construction method,two classes of LDPC codes with column weight 2,whose girths are respectively 16 and 24,can be obtained.The simulation studies show that these codes can achieve better performance than randomly constructed regular LDPC codes over AWGN channels with iterative decoding.At the BER of 10-5,the new proposed(3,6)-regular(4356,2205) QC-LDPC code performs 1.48dB from the Shannon limit.Moreover,it outperforms the random regular LDPC code of comparable parameters by 0.1dB.For the new proposed structured LDPC codes with column weight 2,they can achieve about 2dB coding gains with respect to the random regular LDPC codes with similar block lengths and code rates.
This paper presents a low-complexity recursive and systematic method to construct good well-structured low-density parity-check (LDPC) codes. The method is based on a recursive application of a partial Kronecker product operation on a given gamma x q, q ges 3 a prime, integer lattice L(gamma x q). The (n - 1)- fold product of L(gamma x q) by itself, denoted Ln(gamma x q), represents a regular quasi-cyclic (QC) LDPC code, denoted (see PDF), of high rate and girth 6. The minimum distance of (see PDF) is equal to that of the core code (see PDF) introduced by L(gamma x q). The support of the minimum weight codewords in (see PDF) are characterized by the support of the same type of codewords in (see PDF). From performance perspective the constructed codes compete with the pseudorandom LDPC codes.