Design and Analysis of Time-Invariant SC-LDPC Convolutional Codes With Small Constraint Length

In this paper, we deal with time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs). Classic design approaches usually start from quasi-cyclic low-density parity-check (QC-LDPC) block codes and exploit suitable unwrapping procedures to obtain SC-LDPC-CCs. We show that the direct design of the SC-LDPC-CCs syndrome former matrix or, equivalently, the symbolic parity-check matrix, leads to codes with smaller syndrome former constraint lengths with respect to the best solutions available in the literature. We provide theoretical lower bounds on the syndrome former constraint length for the most relevant families of SC-LDPC-CCs, under constraints on the minimum length of cycles in their Tanner graphs. We also propose new code design techniques that approach or achieve such theoretical limits.Comment: 30 pages, 5 figures, accepted for publication in IEEE Transactions on Communication

Compact QC-LDPC Block and SC-LDPC Convolutional Codes for Low-Latency Communications

Low decoding latency and complexity are two important requirements of channel codes used in many applications, like machine-to-machine communications. In this paper, we show how these requirements can be fulfilled by using some special quasi-cyclic low-density parity-check block codes and spatially coupled low-density parity-check convolutional codes that we denote as compact. They are defined by parity-check matrices designed according to a recent approach based on sequentially multiplied columns. This method allows obtaining codes with girth up to 12. Many numerical examples of practical codes are provided.Comment: 5 pages, 1 figure, presented at IEEE PIMRC 201

Efficient Search of Compact QC-LDPC and SC-LDPC Convolutional Codes with Large Girth

We propose a low-complexity method to find quasi-cyclic low-density parity-check block codes with girth 10 or 12 and shorter length than those designed through classical approaches. The method is extended to time-invariant spatially coupled low-density parity-check convolutional codes, permitting to achieve small syndrome former constraint lengths. Several numerical examples are given to show its effectiveness.Comment: 4 pages, 3 figures, 1 table, accepted for publication in IEEE Communications Letter

Integer Ring Sieve (IRS) for Constructing Compact QC-LDPC Codes with Large Girth

This paper proposes a new method of construction of compact fully-connected Quasi-Cyclic Low Density Parity Check (QC-LDPC) code with girth g = 10 and g = 12. The originality of the proposed method is to impose constraint on the exponent matrix P to reduce the search space drastically. For a targeted expansion factor of N, the first step of the method is to sieve the integer ring Z_N to make a particular subgroup with specific properties to construct the second column of P (the first column being filled with zeros). The remaining columns of P are determined recursively as multiples of the second column thanks to an adaptation of the sequentially multiplied column (SMC) method where a controlled greedy search is applied at each step. The codes constructed with the proposed semi-algebraic method have lengths that can be significantly shorter than the best counterparts in the literature. To illustrate the great potential of the SMC method, we give the explicit construction of a rate 0.75 irregular LDPC code of size 65, 220 that allows a gain of 0.15 dB compared to the code of same rate and size 64,800 of the DVB-S2

Integer Ring Sieve for Constructing Compact QC-LDPC Codes with Girths 8, 10, and 12

International audienceThis paper proposes a new method of constructing compact fully-connected Quasi-Cyclic Low Density Parity Check (QC-LDPC) codes with girth g = 8, 10, and 12. The originality of the proposed method is to impose constraints on the exponent matrix P to reduce the search space drastically. For a targeted lifting degree of N , the first step of the method is to sieve the integer ring ZN to make a particular subgroup with specific properties to construct the second column of P (the first column being filled with zeros). The remaining columns of P are determined recursively as multiples of the second column by adapting the sequentially multiplied column (SMC) method whereby a controlled greedy search is applied at each step. The codes constructed with the proposed semi-algebraic method show lengths that can be significantly shorter than their best counterparts in the literature

An End-to-End Scheme for Learning Over Compressed Data Transmitted Through a Noisy Channel

International audienceWithin the emerging area of goal-oriented communications, this paper introduces a novel endto-end transmission scheme dedicated to learning over a noisy channel, under the constraint that no prior training dataset is available. In this scheme, the transmitter makes use of powerful Spherical Harmonic Transform and Irregular Hexagonal Quadratic Amplitude Modulation techniques, while the receiver relies on a Complex-Valued Neural Network (CVNN) so as to realize the learning task onto the received noisy data. As a main feature of the proposed scheme, the transmitter is fixed and does not depend on the source statistics, while the receiver is trained from a first data transmission phase, thus providing an efficient transmission-versus-learning approach under the considered constraint. The proposed transmission scheme may be adapted to a variety of learning problems, and the paper specifically investigates clustering and classification, two very common learning tasks. In the last part of the paper, the source/channel coding rate of the proposed transmission scheme is evaluated theoretically and from numerical simulations. This analysis shows a clear advantage in terms of coding rate of our scheme compared to conventional coding approaches, when targeting the same learning performance level

Efficient Search of Girth-Optimal QC-LDPC Codes

In this paper, we study the cycle structure of quasi-cyclic (QC) low-density parity-check (LDPC) codes with the goal of obtaining the shortest code with a given degree distribution and girth. We focus on QC-LDPC codes, whose Tanner graphs are cyclic liftings of fully connected base graphs of size 3 × n, n 4, and obtain minimal lifting degrees that result in girths 6 and 8. This is performed through an efficient exhaustive search, and as a result, we also find all the possible non-isomorphic codes with the same minimum block length, girth, and degree distribution. The exhaustive search, which is ordinarily a formidable task, is made possible by pruning the search space of many codes that are isomorphic to those previously examined in the search process. Many of the pruning techniques proposed in this paper are also applicable to QC-LDPC codes with base graphs other than the 3 × n fully connected ones discussed here, as well as to codes with a larger girth. To further demonstrate the effectiveness of the pruning techniques, we use them to search for QC-LDPC codes with girths 10 and 12, and find a number of such codes that have a shorter block length compared with the best known similar codes in the literature. In addition, motivated by the exhaustive search results, we tighten the lower bound on the block length of QC-LDPC codes of girth 6 constructed from fully connected 3 × n base graphs, and construct codes that achieve the lower bound for an arbitrary value of n 4

Symmetrical Constructions for Regular Girth-8 QC-LDPC Codes

In this paper, we propose new constructions for regular girth-8 quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices (CPM). The constructions assume symmetries in the structure of the parity-check matrix and employ a greedy exhaustive search algorithm to find the permutation shifts of the CPMs. As a result of symmetries, the new codes have a more compact representation compared with their counterparts. In majority of cases, also, they achieve the girth 8 at a shorter block length for the same degree distribution (code rate). Deterministic (explicit) constructions are also presented to expand the proposed parity-check matrices to larger block lengths and higher rates. The proposed long high-rate codes are often substantially shorter than regular girth-8 QC-LDPC codes of similar rate in the literature. Simulation results demonstrate that the proposed symmetric codes have competitive performance in comparison with similar existing QC-LDPC codes that lack symmetry

Efficient search of compact QC-LDPC and SC-LDPC convolutional codes with large girth

We propose a low-complexity method to find quasi-cyclic low-density parity-check block codes with girth 10 or 12 and length shorter than those designed through classical approaches. The method is extended to time-invariant spatially coupled low-density parity-check convolutional codes, permitting to achieve small syndrome former constraint lengths. Several numerical examples are given to show its effectiveness