Stopping Set Distributions of Some Linear Codes

Stopping sets and stopping set distribution of an low-density parity-check code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let $C$ be a binary $[n,k]$ linear code with parity-check matrix $H$, where the rows of $H$ may be dependent. A stopping set $S$ of $C$ with parity-check matrix $H$ is a subset of column indices of $H$ such that the restriction of $H$ to $S$ does not contain a row of weight one. The stopping set distribution $\{T_i(H)\}_{i=0}^n$ enumerates the number of stopping sets with size $i$ of $C$ with parity-check matrix $H$. Note that stopping sets and stopping set distribution are related to the parity-check matrix $H$ of $C$. Let $H^{*}$ be the parity-check matrix of $C$ which is formed by all the non-zero codewords of its dual code $C^{\perp}$. A parity-check matrix $H$ is called BEC-optimal if $T_i(H)=T_i(H^*), i=0,1,..., n$ and $H$ has the smallest number of rows. On the BEC, iterative decoder of $C$ with BEC-optimal parity-check matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. 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.Comment: 33 pages, submitted to IEEE Trans. Inform. Theory, Feb. 201

Stopping Sets of Algebraic Geometry Codes

Abstract — Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let C be an [n, k] linear code over Fq with parity-check matrix H, wheretherowsof H may be dependent. Let [n] ={1, 2,...,n} denote the set of column indices of H. Astopping set S of C with parity-check matrix H is a subset of [n] such that the restriction of H to S does not contain a row of weight 1. The stopping set distribution {Ti (H)} n i=0 enumerates the number of stopping sets with size i of C with parity-check matrix H. Denote H ∗ , the paritycheck matrix, consisting of all the nonzero codewords in the dual code C ⊥. In this paper, we study stopping sets and stopping set distributions of some residue algebraic geometry (AG) codes with parity-check matrix H ∗. First, we give two descriptions of stopping sets of residue AG codes. For the simplest AG codes, i.e., the generalized Reed–Solomon codes, it is easy to determine all the stopping sets. Then, we consider the AG codes from elliptic curves. We use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets. Then, the stopping sets, the stopping set distribution, and the stopping distance of the AG code from an elliptic curve are reduced to the search, counting, and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively. Finally, for some special cases, we determine the stopping set distributions of the AG codes from elliptic curves. Index Terms — Algebraic geometry codes, elliptic curves, stopping distance, stopping sets, stopping set distribution, subset sum problem. I

A Deterministic Algorithm for Computing the Weight Distribution of Polar Codes

We present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distributions that arise in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, we can compute the entire weight distribution of certain polar cosets in time O(n^2). Any polar code can be represented as a disjoint union of such cosets; moreover, this representation extends to polar codes with dynamically frozen bits. This implies that our methods can be also used to compute the weight distribution of polar codes with CRC precoding, of polarization-adjusted convolutional (PAC) codes and, in fact, general linear codes. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the mixing factor. To reduce the exponential complexity of our algorithm, we make use of the fact that polar codes have a large automorphism group, which includes the lower-triangular affine group LTA(m,2). We prove that LTA(m,2) acts transitively on certain sets of monomials, thereby drastically reducing the number of polar cosets we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of any polar code of length up to n=128

Decoder-in-the-Loop: Genetic Optimization-based LDPC Code Design

LDPC code design tools typically rely on asymptotic code behavior and are affected by an unavoidable performance degradation due to model imperfections in the short length regime. We propose an LDPC code design scheme based on an evolutionary algorithm, the Genetic Algorithm (GenAlg), implementing a "decoder-in-the-loop" concept. It inherently takes into consideration the channel, code length and the number of iterations while optimizing the error-rate of the actual decoder hardware architecture. We construct short length LDPC codes (i.e., the parity-check matrix) with error-rate performance comparable to, or even outperforming that of well-designed standardized short length LDPC codes over both AWGN and Rayleigh fading channels. Our proposed algorithm can be used to design LDPC codes with special graph structures (e.g., accumulator-based codes) to facilitate the encoding step, or to satisfy any other practical requirement. Moreover, GenAlg can be used to design LDPC codes with the aim of reducing decoding latency and complexity, leading to coding gains of up to $0.325$ dB and $0.8$ dB at BLER of $10^{-5}$ for both AWGN and Rayleigh fading channels, respectively, when compared to state-of-the-art short LDPC codes. Also, we analyze what can be learned from the resulting codes and, as such, the GenAlg particularly highlights design paradigms of short length LDPC codes (e.g., codes with degree-1 variable nodes obtain very good results).Comment: in IEEE Access, 201

Construction of Capacity-Achieving Lattice Codes: Polar Lattices

In this paper, we propose a new class of lattices constructed from polar codes, namely polar lattices, to achieve the capacity \frac{1}{2}\log(1+\SNR) of the additive white Gaussian-noise (AWGN) channel. Our construction follows the multilevel approach of Forney \textit{et al.}, where we construct a capacity-achieving polar code on each level. The component polar codes are shown to be naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We prove that polar lattices are \emph{AWGN-good}. Furthermore, using the technique of source polarization, we propose discrete Gaussian shaping over the polar lattice to satisfy the power constraint. Both the construction and shaping are explicit, and the overall complexity of encoding and decoding is $O(N\log N)$ for any fixed target error probability.Comment: full version of the paper to appear in IEEE Trans. Communication