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European Transactions on Telecommunications

Regular low-density parity-check codes from oval designs

Steven R. Weller, Sarah J. Johnson

Published Jan 1, 2003• 22 citations
Publisher

Abstract

This paper presents a construction of low-density parity-check (LDPC) codes based on the incidence matrices of oval designs. The new LDPC codes have regular parity-check matrices and Tanner graphs free of 4-cycles. Like the finite geometry codes, the codes from oval designs have parity-check matrices with a large proportion of linearly dependent rows and can achieve significantly better minimum distances than equivalent length and rate randomly constructed LDPC codes. Further, by exploiting the resolvability of oval designs, and also by employing column splitting, we are able to produce 4-cycle free LDPC codes for a wide range of code rates and lengths while maintaining code regularity. Copyright © 2003 AEI.

DOI
10.1002/ett.4460140503

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This paper presents a construction of low-density parity-check (LDPC) codes based on the incidence matrices of oval designs. The new LDPC codes have regular parity-check matrices and Tanner graphs free of 4-cycles. Like the finite geometry codes, the codes from oval designs have parity-check matrices with a large proportion of linearly dependent rows and can achieve significantly better minimum distances than equivalent length and rate randomly constructed LDPC codes. Further, by exploiting the resolvability of oval designs, and also by employing column splitting, we are able to produce 4-cycle free LDPC codes for a wide range of code rates and lengths while maintaining code regularity. Copyright # 2003 AEI. 1.

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This paper presents a construction of low-density parity-check (LDPC) codes based on the incidence matrices of oval designs. The new LDPC codes have regular parity-check matrices and Tanner graphs free of 4-cycles. Like the finite geometry codes, the codes from oval designs have parity-check matrices with a large proportion of linearly dependent rows and can achieve significantly better minimum distances than equivalent length and rate randomly constructed LDPC codes. Further, by exploiting the resolvability of oval designs, and also by employing column splitting, we are able to produce 4-cycle free LDPC codes for a wide range of code rates and lengths while maintaining code regularity. Copyright © 2003 AEI.
This paper presents a construction of low-density parity-check (LDPC) codes based on the incidence matrices of oval designs. The new LDPC codes have regular parity-check matrices and Tanner graphs free of 4-cycles. Like the finite geometry codes, the codes from oval designs have parity-check matrices with a large proportion of linearly dependent rows and can achieve significantly better minimum distances than equivalent length and rate randomly constructed LDPC codes. Further, by exploiting the resolvability of oval designs, and also by employing column splitting, we are able to produce 4-cycle free LDPC codes for a wide range of code rates and lengths while maintaining code regularity. Copyright # 2003 AEI. 1.
The paper presents a construction of very high-rate low-density parity-check (LDPC) codes based on incidence matrices of unital designs. Like the projective geometry and oval designs, unital designs exist with incidence matrices which are significantly rank deficient. Thus high-rate LDPC codes with a large number of linearly dependent parity-check equations can be constructed. The LDPC codes from unitals have Tanner graphs free of 4-cycles and perform well with iterative decoding, offering new LDPC codes at rates and lengths not available with existing algebraic LDPC codes.
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.
A new approach is proposed for constructing regular low-density parity-check (LDPC) codes based on tensor product of matrices. In this paper, first a general construction method of regular LDPC codes exploiting permutation matrices is described. Constructed codes have a quasi-cyclic structure with no short cycles of length 4 in their Tanner graph, hence simple encoding while maintaining good performance is achieved. The paper also demonstrates a generalized design, which covers a large family of LDPC codes and number of other construction methods. The new generalized LDPC codes are defined by a small number of parameters and cover a large set of code lengths and rates. Using these codes, LDPC matrices of any column weight and row weight can be constructed. Performance of these codes under iterative decoding compares well with other well-structured as well as random LDPC codes
This paper gives a new random construction for LDPC codes, which uses an iterative algorithm to avoid short cycles in the Tanner graph. The construction method has low complexity and great flexible choice in LDPC code parameters including code length, code rate, least girth of the graph, and the weight of column and row in the parity-check matrix. The method can be applied to the irregular LDPC codes and strict regular LDPC codes. Simulations show that the method performs well with iterative decoding.