On the pseudocodeword weight and parity-check matrix redundancy of linear codes

In this letter, the stopping sets and stopping distance of finite geometry LDPC (FG-LDPC) codes are studied. It is known that FG-LDPC codes are majority-logic decodable and a lower bound on the minimum distance can be thus obtained. It is shown in this letter that this lower bound on the minimum distance of FG-LDPC codes is also a lower bound on the stopping distance of FG-LDPC codes, which implies that FG-LDPC codes have considerably large stopping distance. This may explain in one respect why some FG-LDPC codes perform well with iterative decoding in spite of having many cycles of length 4 in their Tanner graphs.

Stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> be a binary [ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ] linear code with parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> , where the rows of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> may be dependent. A stopping set <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> with parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> is a subset of column indices of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> such that the restriction of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> does not contain a row of weight one. The stopping set distribution { <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ti</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> )} <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> =0 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> enumerates the number of stopping sets with size <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> with parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> . Note that stopping sets and stopping set distribution are related to the parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> . Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> be the parity-check matrix of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> which is formed by all the nonzero codewords of its dual code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">⊥</sup> . A parity-check matrix <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> is called BEC-optimal if <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ti</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> )= <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ti</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> ), <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> =0,1,..., <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> has the smallest number of rows. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes, and the extended Hamming codes, which are some Reed-Muller codes or their shortening or puncturing versions.

For a binary linear code, the pseudocodeword redundancy with respect to the additive white Gaussian noise channel, the binary symmetric channel, or the max-fractional weight is defined to be the smallest number of rows in a parity-check matrix such that the corresponding minimum pseudoweight is equal to the minimum Hamming distance of the code. It is shown that most codes do not have a finite pseudocodeword redundancy. Also, upper bounds on the pseudocodeword redundancy for some families of codes, including codes based on designs, are provided. The pseudocodeword redundancies for all codes of small length (at most 9) are computed. Furthermore, comprehensive results are provided on the cases of cyclic codes of length at most 250 for which the eigenvalue bound of Vontobel and Koetter is sharp.

We present a novel iterative decoding algorithm for Reed-Muller (RM) codes, which takes advantage of a graph representation of the code. Vertices of the considered graph correspond to codewords, with two vertices being connected by an edge if and only if the Hamming distance between the corresponding codewords equals the minimum distance of the code. The algorithm uses a greedy local search to find a node optimizing a metric, e.g. the correlation between the received vector and the corresponding codeword. In addition, the cyclic redundancy check can be used to terminate the search as soon as a valid codeword is found, leading to an improvement in the average computational complexity of the algorithm. Simulation results for both binary symmetric channel and additive white Gaussian noise channel show that the presented decoder approaches the performance of maximum likelihood decoding for RM codes of length less than 1024 and for the second-order RM codes of length less than 4096. Moreover, it is demonstrated that the considered decoding approach outperforms state-of-the-art decoding algorithms of RM codes with similar computational complexity for a wide range of block lengths and rates.

In this paper, we re-introduce from our previous work [1] a new family of nonlinear codes, called weak flip codes, and show that its subfamily fair weak flip codes belongs to the class of equidistant codes, satisfying that any two distinct codewords have identical Hamming distance. It is then noted that the fair weak flip codes are related to the binary nonlinear Hadamard codes as both code families maximize the minimum Hamming distance and meet the Plotkin upper bound under certain blocklengths. Although the fair weak flip codes have the largest minimum Hamming distance and achieve the Plotkin bound, we find that these codes are by no means optimal in the sense of average error probability over binary symmetric channels (BSC). In parallel, this result implies that the equidistant Hadamard codes are also nonoptimal over BSCs. Such finding is in contrast to the conventional code design that aims at the maximization of the minimum Hamming distance. The results in this paper are proved by examining the exact error probabilities of these codes on BSCs, using the column-wise analysis on the codebook matrix.

Reed-Muller (RM) codes and polar codes are generated by the same matrix $G_m= \bigl[\begin{smallmatrix}1 & 0 \\ 1 & 1 \\ \end{smallmatrix}\bigr]^{\otimes m}$ but using different subset of rows. RM codes select simply rows having largest weights. Polar codes select instead rows having the largest conditional mutual information proceeding top to down in $G_m$; while this is a more elaborate and channel-dependent rule, the top-to-down ordering has the advantage of making the conditional mutual information polarize, giving directly a capacity-achieving code on any binary memoryless symmetric channel (BMSC). RM codes are yet to be proved to have such property. In this paper, we reconnect RM codes to polarization theory. It is shown that proceeding in the RM code ordering, i.e., not top-to-down but from the lightest to the heaviest rows in $G_m$, the conditional mutual information again polarizes. We further demonstrate that it does so faster than for polar codes. This implies that $G_m$ contains another code, different than the polar code and called here the twin code, that is provably capacity-achieving on any BMSC. This proves a necessary condition for RM codes to achieve capacity on BMSCs. It further gives a sufficient condition if the rows with the largest conditional mutual information correspond to the heaviest rows, i.e., if the twin code is the RM code. We show here that the two codes bare similarity with each other and give further evidence that they are likely the same.

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

Most decoding algorithms of linear codes, in general, are designed to correct or detect errors. However, many channels cause erasures in addition to errors. In principle, decoding over such channels can be accomplished by deleting the erased symbols and decoding the resulting vector with respect to a punctured code. For any given linear code and any given maximum number of correctable erasures, parity-check matrices are introduced that yield parity-check equations which do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures. These matrices allow for the separation of erasures from errors to facilitate decoding. Several constructions of such separating parity-check matrices are presented. To reduce decoding complexity, separating parity-check matrices with small number of rows are preferred. The minimum number of rows in a parity-check matrix separating a given maximum number of erasures is called the separating redundancy. Upper and lower bounds on the separating redundancies are derived. In particular, it is shown that the separating redundancies tend to grow linearly with the number of rows in full-rank parity-check matrices of codes. The separating redundancies of some classes of codes are determined for some maximum numbers of erasures.

On the binary erasure channel (BEC), the performance of low-density parity-check (LDPC) codes decoded by iterative algorithms is estimated by the small stopping sets in the codes. The size of smallest stopping set is called the stopping distance. Since it depends on the structure of parity-check matrix for the code, we can construct the parity-check matrix by adding dependent rows so that the stopping distance is equal to the minimum distance. The stopping redundancy is defined as the smallest number of rows in such a parity-check matrix. In this paper, we propose a probabilistic algorithm for computing the stopping redundancy of LDPC codes. Additionally, we show numerical results of computing the stopping redundancy of (155,64), (504,252) and (1008,504) LDPC codes.

Reed Muller (RM) codes are known for their good minimum distance. One can use their structure to construct polar-like codes with good distance properties by choosing the information set as the rows of the polarization matrix with the highest Hamming weight, instead of the most reliable synthetic channels. However, the information length options of RM codes are quite limited due to their specific structure. In this work, we present sufficient conditions to increase the information length by at least one bit for some underlying RM codes and in order to obtain pre-transformed polar-like codes with the same minimum distance than lower rate codes. Moreover, our findings are combined with the method presented in [1] to further reduce the number of minimum weight codewords. Numerical results show that the designed codes perform close to the meta-converse bound at short blocklengths and better than the polarized adjusted convolutional polar codes with the same parameters.

Among the classes of LDPC codes that have been constructed and designed, the only class of LDPC codes that are cyclic is the class of codes constructed based on the incidence vectors of lines of finite geometries, called finite geometry LDPC codes. Cyclic finite geometry LDPC codes are known to have large minimum distances and can provide good error performance with very low error-floors using iterative decoding based on belief propagation. Their cyclic structure allows them to be efficiently and systematically encoded with simple shift-registers in linear time with linear complexity. An obvious question is whether, besides cyclic finite geometry LDPC codes, there are other cyclic codes with large minimum distances that can be efficiently decoded iteratively using channel soft information. In this paper, we present one such class of cyclic codes. Codes in this class are two-step majority-logic decodable and they are also constructed based on finite geometries. Two iterative decoding algorithms are devised for this class of cyclic codes and they provide significant coding gain over the two-step majority-logic decoding of codes in this class.

Agent-based models are nowadays widely used; however, their calibration on real data still remains an open issue which prevents to exploit completely their potentiality. Rarely such a kind of models can be studied analytically; more often they are studied by simulation. Among the problems encountered in ABM calibration, the choice of the criteria to fit can appear arbitrary. Markov chain analysis can come through to identify a standard procedure able to face this issue. Indeed, Izquierdo et al. (J Artif Soc Soc Simul 12(16):1–6, 2009) show that many computer simulation models can be represented as Markov chains. Exploiting such an idea classical minimum distance and its simulated counterpart, i.e., simulated minimum distance, are discussed theoretically and applied to Kirman model, which can be reformulated as a Markov chain. Comparison with approximate Bayesian computation is also addressed.

In this paper, we propose a binary message-passing algorithm for decoding low-density parity-check (LDPC) codes. The algorithm substantially improves the performance of purely hard-decision iterative algorithms with a small increase in the memory requirements and the computational complexity. We associate a reliability value to each nonzero element of the code's parity-check matrix, and differentially modify this value in each iteration based on the sum of the extrinsic binary messages from the check nodes. For the tested random and finite-geometry LDPC codes, the proposed algorithm can achieve performance as close as 1.3 dB and 0.7 dB to that of belief propagation (BP) at the error rates of interest, respectively. This is while, unlike BP, the algorithm does not require the estimation of channel signal to noise ratio. Low memory and computational requirements and binary message-passing make the proposed algorithm attractive for high-speed low-power applications.

In this paper, we propose a binary message-passing algorithm for decoding low-density parity-check (LDPC) codes. The algorithm substantially improves the performance of purely hard-decision iterative algorithms with a small increase in the memory requirements and the computational complexity. We associate a reliability value to each nonzero element of the code's parity-check matrix, and differentially modify this value in each iteration based on the sum of the extrinsic binary messages from the check nodes. For the tested random and finitegeometry LDPC codes, the proposed algorithm can perform as close as about 1 dB and 0.5 dB to belief propagation (BP) at the error rates of interest, respectively. This is while, unlike BP, the algorithm does not require the estimation of channel signal to noise ratio. Low memory and computational requirements and binary message-passing make the proposed algorithm attractive for high-speed low-power applications.

This paper present an algebraic and systematic method for constructing low density parity-check (LDPC) codes from a geometric point of view. LDPC codes are constructed based on the lines and points of a finite geometry. Two classes of LDPC codes are constructed based on the well known Euclidean and projective geometries over finite fields. Codes of these two classes are cyclic and they have good minimum distances and structural properties. They can be decoded in various ways and perform very well with the iterative decoding based on belief propagation (IDBP). These finite geometry LDPC codes can be extended and shortened in various ways. Extension by splitting columns of the parity-check matrices of these codes results in good extended finite geometry LDPC codes. Long extended finite geometry LDPC codes have been constructed and they achieve an error performance only a few tenths of a dB away from the Shannon limit with IDBP. Finite geometry LDPC codes are strong competitors to turbo codes for error control in communication and digital data storage systems.