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

Linear Transformations for Randomness Extraction

Information-efficient approaches for extracting randomness from imperfect sources have been extensively studied, but simpler and faster ones are required in the high-speed applications of random number generation. In this paper, we focus on linear constructions, namely, applying linear transformation for randomness extraction. We show that linear transformations based on sparse random matrices are asymptotically optimal to extract randomness from independent sources and bit-fixing sources, and they are efficient (may not be optimal) to extract randomness from hidden Markov sources. Further study demonstrates the flexibility of such constructions on source models as well as their excellent information-preserving capabilities. Since linear transformations based on sparse random matrices are computationally fast and can be easy to implement using hardware like FPGAs, they are very attractive in the high-speed applications. In addition, we explore explicit constructions of transformation matrices. We show that the generator matrices of primitive BCH codes are good choices, but linear transformations based on such matrices require more computational time due to their high densities.Comment: 2 columns, 14 page

On generalized LDPC codes for ultra reliable communication

Ultra reliable low latency communication (URLLC) is an important feature in future mobile communication systems, as they will require high data rates, large system capacity and massive device connectivity [11]. To meet such stringent requirements, many error-correction codes (ECC)s are being investigated; turbo codes, low density parity check (LDPC) codes, polar codes and convolutional codes [70, 92, 38], among many others. In this work, we present generalized low density parity check (GLDPC) codes as a promising candidate for URLLC. Our proposal is based on a novel class of GLDPC code ensembles, for which new analysis tools are proposed. We analyze the trade-o_ between coding rate and asymptotic performance of a class of GLDPC codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. To incorporate both bounded distance (BD) and maximum likelihood (ML) decoding at GC nodes into our analysis without resorting to multi-edge type of degree distribution (DD)s, we propose the probabilistic peeling decoding (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a successful decoding probability that depends on both the GC block code as well as its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as Density evolution (DE) or the differential equation method. We demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code under ML decoding at GC nodes. We illustrate our analysis for GLDPC code ensembles with regular and irregular DDs. This design methodology is applied to construct practical codes for URLLC. To this end, we incorporate to our analysis the use of quasi-cyclic (QC) structures, to mitigate the code error floor and facilitate the code very large scale integration (VLSI) implementation. Furthermore, for the additive white Gaussian noise (AWGN) channel, we analyze the complexity and performance of the message passing decoder with various update rules (including standard full-precision sum product and min-sum algorithms) and quantization schemes. The block error rate (BLER) performance of the proposed GLDPC codes, combined with a complementary outer code, is shown to outperform a variety of state-of-the-art codes, for URLLC, including LDPC codes, polar codes, turbo codes and convolutional codes, at similar complexity rates.Programa Oficial de Doctorado en Multimedia y ComunicacionesPresidente: Juan José Murillo Fuentes.- Secretario: Matilde Pilar Sánchez Fernández.- Vocal: Javier Valls Coquilla

Channel Coding at Low Capacity

Low-capacity scenarios have become increasingly important in the technology of the Internet of Things (IoT) and the next generation of mobile networks. Such scenarios require efficient and reliable transmission of information over channels with an extremely small capacity. Within these constraints, the performance of state-of-the-art coding techniques is far from optimal in terms of either rate or complexity. Moreover, the current non-asymptotic laws of optimal channel coding provide inaccurate predictions for coding in the low-capacity regime. In this paper, we provide the first comprehensive study of channel coding in the low-capacity regime. We will investigate the fundamental non-asymptotic limits for channel coding as well as challenges that must be overcome for efficient code design in low-capacity scenarios.Comment: 39 pages, 5 figure

Quantum Random Access Codes with Shared Randomness

We consider a communication method, where the sender encodes n classical bits into 1 qubit and sends it to the receiver who performs a certain measurement depending on which of the initial bits must be recovered. This procedure is called (n,1,p) quantum random access code (QRAC) where p > 1/2 is its success probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not possible. We extend this model with shared randomness (SR) that is accessible to both parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any n > 0. We give an upper bound on its success probability (the known (2,1,0.85) and (3,1,0.79) QRACs match this upper bound). We discuss some particular constructions for several small values of n. We also study the classical counterpart of this model where n bits are encoded into 1 bit instead of 1 qubit and SR is used. We give an optimal construction for such codes and find their success probability exactly--it is less than in the quantum case. Interactive 3D quantum random access codes are available on-line at http://home.lanet.lv/~sd20008/racs .Comment: 51 pages, 33 figures. New sections added: 1.2, 3.5, 3.8.2, 5.4 (paper appears to be shorter because of smaller margins). Submitted as M.Math thesis at University of Waterloo by M