arXiv subjects

Mark F. Flanagan

Publications and source records attributed to Mark F. Flanagan.

85 records · Page 5Linked to original sources

arXiv2010

On the Growth Rate of the Weight Distribution of Irregular Doubly-Generalized LDPC Codes

In this paper, an expression for the asymptotic growth rate of the number of small linear-weight codewords of irregular doubly-generalized LDPC (D-GLDPC) codes is derived. The expression is compact and generalizes existing results for LDPC and generalized LDPC (GLDPC) codes. Ensembles with check or variable node minimum distance greater than 2 are shown to be have good growth rate behavior, while for other ensembles a fundamental parameter is identified which discriminates between an asymptotically small and an asymptotically large expected number of small linear-weight codewords. Also, in the latter case it is shown that the growth rate depends only on the check and variable nodes with minimum distance 2. An important connection between this new result and the stability condition of D-GLDPC codes over the BEC is highlighted. Such a connection, previously observed for LDPC and GLDPC codes, is now extended to the case of D-GLDPC codes. Finally, it is shown that the analysis may be extended to include the growth rate of the stopping set size distribution of irregular D-GLDPC codes.

cs.IT↗

arXiv2010

On the Pseudocodeword Redundancy

We define the AWGNC, BSC, and max-fractional pseudocodeword redundancy of a code as the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance. We show that most codes do not have a finite pseudocodeword redundancy. We also provide bounds on the pseudocodeword redundancy for some families of codes, including codes based on designs.

cs.IT↗

arXiv2009

Spectral Graph Analysis of Quasi-Cyclic Codes

In this paper we analyze the bound on the additive white Gaussian noise channel (AWGNC) pseudo-weight of a (c,d)-regular linear block code based on the two largest eigenvalues of H^T H. In particular, we analyze (c,d)-regular quasi-cyclic (QC) codes of length rL described by J x L block parity-check matrices with circulant block entries of size r x r. We proceed by showing how the problem of computing the eigenvalues of the rL x rL matrix H^T H can be reduced to the problem of computing eigenvalues for r matrices of size L x L. We also give a necessary condition for the bound to be attained for a circulant matrix H and show a few classes of cyclic codes satisfying this criterion.

cs.IT↗

arXiv2009

Exposing Pseudoweight Layers in Regular LDPC Code Ensembles

A solution is presented for the asymptotic growth rate of the AWGN-pseudoweight distribution of regular low-density parity-check (LDPC) code ensembles for a selected graph cover degree M >= 1. The evaluation of the growth rate requires solution of a system of 2M+1 nonlinear equations in 2M+1 unknowns. Simulation results for the pseudoweight distribution of two regular LDPC code ensembles are presented for graph covers of low degree.

cs.IT↗

arXiv2009

Growth Rate of the Weight Distribution of Doubly-Generalized LDPC Codes: General Case and Efficient Evaluation

The growth rate of the weight distribution of irregular doubly-generalized LDPC (D-GLDPC) codes is developed and in the process, a new efficient numerical technique for its evaluation is presented. The solution involves simultaneous solution of a 4 x 4 system of polynomial equations. This represents the first efficient numerical technique for exact evaluation of the growth rate, even for LDPC codes. The technique is applied to two example D-GLDPC code ensembles.

cs.IT↗

arXiv2009

Linear-Programming Decoding of Nonbinary Linear Codes

A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates linear-programming based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the 'maximum-likelihood certificate' property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. Two alternative polytopes for use with linear-programming decoding are studied, and it is shown that for many classes of codes these polytopes yield a complexity advantage for decoding. These polytope representations lead to polynomial-time decoders for a wide variety of classical nonbinary linear codes. LP decoding performance is illustrated for the [11,6] ternary Golay code with ternary PSK modulation over AWGN, and in this case it is shown that the performance of the LP decoder is comparable to codeword-error-rate-optimum hard-decision based decoding. LP decoding is also simulated for medium-length ternary and quaternary LDPC codes with corresponding PSK modulations over AWGN.

cs.IT↗

arXiv2008

Linear-Programming Receivers

It is shown that any communication system which admits a sum-product (SP) receiver also admits a corresponding linear-programming (LP) receiver. The two receivers have a relationship defined by the local structure of the underlying graphical model, and are inhibited by the same phenomenon, which we call 'pseudoconfigurations'. This concept is a generalization of the concept of 'pseudocodewords' for linear codes. It is proved that the LP receiver has the 'optimum certificate' property, and that the receiver output is the lowest cost pseudoconfiguration. Equivalence of graph-cover pseudoconfigurations and linear-programming pseudoconfigurations is also proved. While the LP receiver is generally more complex than the corresponding SP receiver, the LP receiver and its associated pseudoconfiguration structure provide an analytic tool for the analysis of SP receivers. As an example application, we show how the LP design technique may be applied to the problem of joint equalization and decoding.

cs.IT↗

arXiv2008

On the Growth Rate of the Weight Distribution of Irregular Doubly-Generalized LDPC Codes

cs.IT↗

arXiv2008

Lower Bounds on the Minimum Pseudodistance for Linear Codes with $q$-ary PSK Modulation over AWGN

We present lower bounds on the minimum pseudocodeword effective Euclidean distance (or minimum "pseudodistance") for coded modulation systems using linear codes with $q$-ary phase-shift keying (PSK) modulation over the additive white Gaussian noise (AWGN) channel. These bounds apply to both binary and nonbinary coded modulation systems which use direct modulation mapping of coded symbols. The minimum pseudodistance may serve as a first-order measure of error-correcting performance for both linear-programming and message-passing based receivers. In the case of a linear-programming based receiver, the minimum pseudodistance may be used to form an exact bound on the codeword error rate of the system.

cs.IT↗

arXiv2008

Codeword-Independent Performance of Nonbinary Linear Codes Under Linear-Programming and Sum-Product Decoding

A coded modulation system is considered in which nonbinary coded symbols are mapped directly to nonbinary modulation signals. It is proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. It is shown that this result holds for both linear-programming decoders and sum-product decoders. In particular, this provides a natural modulation mapping for nonbinary codes mapped to PSK constellations for transmission over memoryless channels such as AWGN channels or flat fading channels with AWGN.

cs.IT↗

arXiv2008

Polytope Representations for Linear-Programming Decoding of Non-Binary Linear Codes

In previous work, we demonstrated how decoding of a non-binary linear code could be formulated as a linear-programming problem. In this paper, we study different polytopes for use with linear-programming decoding, and show that for many classes of codes these polytopes yield a complexity advantage for decoding. These representations lead to polynomial-time decoders for a wide variety of classical non-binary linear codes.

cs.IT↗

arXiv2008

Combinatorial Gray codes for classes of pattern avoiding permutations

The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, Schröder, Pell, even index Fibonacci numbers and the central binomial coefficients. Consequently, this provides Gray codes for $\s_n(τ)$ for all $τ\in \s_3$ and the obtained Gray codes have distances 4 and 5.

math.CO↗

arXiv2007

Linear-programming Decoding of Non-binary Linear Codes

We develop a framework for linear-programming (LP) decoding of non-binary linear codes over rings. We prove that the resulting LP decoder has the `maximum likelihood certificate' property, and we show that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. LP decoding performance is illustrated for the (11,6,5) ternary Golay code with ternary PSK modulation over AWGN, and in this case it is shown that the LP decoder performance is comparable to codeword-error-rate-optimum hard-decision based decoding.

cs.IT↗