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29
Lowdensity paritycheck codes based on finite geometries: A rediscovery and new results
 IEEE Trans. Inform. Theory
, 2001
"... This paper presents a geometric approach to the construction of lowdensity paritycheck (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and thei ..."
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Cited by 121 (4 self)
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This paper presents a geometric approach to the construction of lowdensity paritycheck (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth T. Finitegeometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasicyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finitegeometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finitegeometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.
Graphcover decoding and finitelength analysis of messagepassing iterative decoding of LDPC codes
 IEEE TRANS. INFORM. THEORY
, 2005
"... The goal of the present paper is the derivation of a framework for the finitelength analysis of messagepassing iterative decoding of lowdensity paritycheck codes. To this end we introduce the concept of graphcover decoding. Whereas in maximumlikelihood decoding all codewords in a code are comp ..."
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Cited by 67 (12 self)
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The goal of the present paper is the derivation of a framework for the finitelength analysis of messagepassing iterative decoding of lowdensity paritycheck codes. To this end we introduce the concept of graphcover decoding. Whereas in maximumlikelihood decoding all codewords in a code are competing to be the best explanation of the received vector, under graphcover decoding all codewords in all finite covers of a Tanner graph representation of the code are competing to be the best explanation. We are interested in graphcover decoding because it is a theoretical tool that can be used to show connections between linear programming decoding and messagepassing iterative decoding. Namely, on the one hand it turns out that graphcover decoding is essentially equivalent to linear programming decoding. On the other hand, because iterative, locally operating decoding algorithms like messagepassing iterative decoding cannot distinguish the underlying Tanner graph from any covering graph, graphcover decoding can serve as a model to explain the behavior of messagepassing iterative decoding. Understanding the behavior of graphcover decoding is tantamount to understanding
Paritycheck density versus performance of binary linear block codes over memoryless symmetric channels
 IEEE Trans. on Information Theory
, 2003
"... Lowdensity paritycheck (LDPC) codes are efficiently encoded and decoded due to the sparseness of their paritycheck matrices. Motivated by their remarkable performance and feasible complexity under iterative messagepassing decoding, we derive lower bounds on the density of paritycheck matrices o ..."
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Cited by 48 (19 self)
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Lowdensity paritycheck (LDPC) codes are efficiently encoded and decoded due to the sparseness of their paritycheck matrices. Motivated by their remarkable performance and feasible complexity under iterative messagepassing decoding, we derive lower bounds on the density of paritycheck matrices of binary linear codes whose transmission takes place over a memoryless binaryinput outputsymmetric (MBIOS) channel. The bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity; they are valid for every sequence of binary linear block codes. For every MBIOS channel, we construct a sequence of ensembles of regular LDPC codes, so that an upper bound on the asymptotic density of their paritycheck matrices scales similarly to the lower bound. The tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of rightregular LDPC codes which was introduced by Shokrollahi, and which is known to achieve the capacity of this channel. Under iterative messagepassing decoding, we show that this sequence of ensembles is asymptotically optimal (in a sense to be defined in this paper), strengthening a result of Shokrollahi. Finally, we derive lower bounds on the bit error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs, and study their performance limitations
Low density parity check codes based on finite geometries: A rediscovery and new results
 IEEE Trans. Inform. Theory
, 2001
"... This paper presents a geometric approach to the construction of low density parity check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and thei ..."
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Cited by 35 (11 self)
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This paper presents a geometric approach to the construction of low density parity check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth 6. Finite geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasicyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a dB away from the Shannon theoretical limit with iterative decoding.
Algorithmic Complexity in Coding Theory and the Minimum Distance Problem
, 1997
"... We start with an overview of algorithmiccomplexity problems in coding theory We then show that the problem of computing the minimum distance of a binary linear code is NPhard, and the corresponding decision problem is NPcomplete. This constitutes a proof of the conjecture Bedekamp, McEliece, van T ..."
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Cited by 34 (2 self)
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We start with an overview of algorithmiccomplexity problems in coding theory We then show that the problem of computing the minimum distance of a binary linear code is NPhard, and the corresponding decision problem is NPcomplete. This constitutes a proof of the conjecture Bedekamp, McEliece, van Tilborg, dating back to 1978. Extensions and applications of this result to other problems in coding theory are discussed.
Resolvable 2Designs for Regular LowDensity ParityCheck Codes
, 2003
"... This paper extends the class of lowdensity paritycheck (LDPC) codes that can be algebraically constructed. We present regular LDPC codes based on resolvable Steiner 2designs which have Tanner graphs free of fourcycles. The resulting codes areregular orregular for any value of and for a flexible ..."
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Cited by 13 (5 self)
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This paper extends the class of lowdensity paritycheck (LDPC) codes that can be algebraically constructed. We present regular LDPC codes based on resolvable Steiner 2designs which have Tanner graphs free of fourcycles. The resulting codes areregular orregular for any value of and for a flexible choice of code lengths.
Which codes have 4cyclefree Tanner graphs
 IEEE Trans. Information Theory
, 2006
"... Abstract — Let C be an [n, k, d] binary linear code with rate R = k/n and dual C ⊥. In this work, it is shown that C can be represented by a 4cyclefree Tanner graph only if: pd ⊥ ≤ $r np(p − 1) + n2 ..."
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Cited by 7 (0 self)
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Abstract — Let C be an [n, k, d] binary linear code with rate R = k/n and dual C ⊥. In this work, it is shown that C can be represented by a 4cyclefree Tanner graph only if: pd ⊥ ≤ $r np(p − 1) + n2
Improved bounds on the paritycheck density and achievable rates of binary linear block codes with applications to LDPC codes,” May 2005, submitted to
 IEEE IT
"... We derive bounds on the asymptotic density of paritycheck matrices and the achievable rates of binary linear block codes transmitted over memoryless binaryinput outputsymmetric (MBIOS) channels. The lower bounds on the density of arbitrary paritycheck matrices are expressed in terms of the gap b ..."
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Cited by 6 (4 self)
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We derive bounds on the asymptotic density of paritycheck matrices and the achievable rates of binary linear block codes transmitted over memoryless binaryinput outputsymmetric (MBIOS) channels. The lower bounds on the density of arbitrary paritycheck matrices are expressed in terms of the gap between the rate of these codes for which reliable communication is achievable and the channel capacity, and the bounds are valid for every sequence of binary linear block codes. These bounds address the question, previously considered by Sason and Urbanke, of how sparse can paritycheck matrices of binary linear block codes be as a function of the gap to capacity. Similarly to a previously reported bound by Sason and Urbanke, the new lower bounds on the paritycheck density scale like the log of the inverse of the gap to capacity, but their tightness is improved (except for a binary symmetric/erasure channel, where they coincide with the previous bound). The new upper bounds on the achievable rates of binary linear block codes tighten previously reported bounds by Burshtein et al., and therefore enable to obtain tighter upper bounds on the thresholds of sequences of binary linear block codes under ML decoding. The bounds are applied to lowdensity paritycheck (LDPC) codes, and the improvement in their tightness is exemplified numerically. The upper bounds on the achievable rates enable to assess the inherent loss in performance of various iterative decoding algorithms as compared to optimal ML decoding. The lower bounds on the asymptotic paritycheck density are helpful in assessing the inherent tradeoff between the asymptotic performance of LDPC codes and their decoding complexity (per iteration) under messagepassing decoding.
Acyclic Tanner Graphs and MaximumLikelihood Decoding of Linear Block Codes
 IEE Proc. Commun
, 1998
"... The maximumlikelihood decoding of linear block codes by the Wagner rule decoding is discussed. In this approach, the Wagner rule decoding which has been primarily applied to single parity check codes is employed on acyclic Tanner graphs. Accordingly, a coset decoding equipped with the Wagner rule d ..."
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Cited by 3 (1 self)
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The maximumlikelihood decoding of linear block codes by the Wagner rule decoding is discussed. In this approach, the Wagner rule decoding which has been primarily applied to single parity check codes is employed on acyclic Tanner graphs. Accordingly, a coset decoding equipped with the Wagner rule decoding is applied to the decoding of a code C having a Tanner graph with cycles. A subcode C 1 of C with acyclic Tanner graph is chosen as the base subcode. All cosets of C 1 have the same Tanner graph and are distinguished by their values of parity nodes in the graph. The acyclic Tanner graph of C 1 together with a trellis representation of the space of the parity sequences represent the code C. Efficient use of this graphical presentation provides a unified and systematic approach to several ever best known maximumlikelihood decoding techniques of linear block codes. The hexacode H 6 , ternary Golay code G 12 , ReedMuller codes, Hamming codes, and the extended quadratic residue codes are discussed.
Bounds on the performance of maximumlikelihood decoded binary block codes in AWGN interference
, 2002
"... I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means ..."
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Cited by 2 (1 self)
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I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. ii The University of Waterloo requires the signatures of all persons using or photocopying this thesis. Please sign below, and give address and date. iii The error probability of MaximumLikelihood (ML) softdecision decoded binary block codes rarely accepts nice closed forms. In addition, for long codes ML decoding becomes prohibitively complex. Nevertheless, bounds on the performance of ML decoded systems provide insight into the effect of system parameters on the