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IEEE Information Theory Workshop (ITW)

Regular low-density parity-check codes from combinatorial designs

Sarah J. Johnson, Steven R. Weller

Published Nov 13, 2002• 78 citations
Publisher

Abstract

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.

DOI
10.1109/itw.2001.955146

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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.
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/.
Short-length codes may play a crucial role in delay-sensitive scenarios. Low-density parity-check (LDPC) codes, as a class of Shannon limit approaching codes, perform well with long code length. However, the design of short-length LDPC codes with good performance is still challenging. This paper investigates the construction of short-length LDPC codes based on the combinatorial design theory. Specifically, Kirkman triple systems (KTS), as a type of balanced incomplete block design (BIBD) techniques, are used to construct the parity-check matrices of LDPC codes, which are guaranteed to be 4-cycles free. Some other properties of such designed LDPC codes are also investigated. For instance, a lower bound on the girth, the code rate and minimum code distance. The simulation results show that the designed short-length LDPC codes based on KTS outperform that of randomly constructed ones.
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.
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.
This paper presents a novel method for constructing structured regular low-density parity-check (LDPC) codes based on a special type of combinatorial designs, known as Steiner systems. This code design approach can be considered as a generalization of the well-known method which uses the point-block incidence matrix of a Steiner 2-design for the code construction. Though the given method can be applied on any Steiner system S(t, k, v), in this paper we focus only on Steiner systems with t ≥ 3. Furthermore, we show that not only a Steiner system (X,B) itself, but also its residual design with respect to an arbitrary point x ∈ X can be employed for code construction. We also present a technique for constructing binary and non-binary QC-LDPC codes from Steiner systems. The Tanner graph of the constructed codes is free of 4-cycles and hence the codes have girth at least six. Simulation results show that the so constructed codes perform well over the AWGN channel with iterative message-passing decoding.