Time-Invariant Spatially Coupled Low-Density Parity-Check Codes with Small Constraint Length

We consider a special family of SC-LDPC codes, that is, time-invariant LDPCC codes, which are known in the literature for a long time. Codes of this kind are usually designed by starting from QC block codes, and applying suitable unwrapping procedures. We show that, by directly designing the LDPCC code syndrome former matrix without the constraints of the underlying QC block code, it is possible to achieve smaller constraint lengths with respect to the best solutions available in the literature. We also find theoretical lower bounds on the syndrome former constraint length for codes with a specified minimum length of the local cycles in their Tanner graphs. For this purpose, we exploit a new approach based on a numerical representation of the syndrome former matrix, which generalizes over a technique we already used to study a special subclass of the codes here considered.Comment: 5 pages, 4 figures, to be presented at IEEE BlackSeaCom 201

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

Structured LDPC convolutional codes

LDPC convolutional codes, also known as spatially coupled LDPC codes, have attracted considerable attention due to their promising properties. By coupling the protographs from different positions into a chain and terminating the chain properly, the resulting convolutional-like LDPC code ensemble is able to produce capacity-achieving performance in the limit of large parameters. In addition, optimization of degree distributions is avoided due to the regularity of the node degrees. Based on these characteristics, we propose in this thesis structured LDPC convolutional codes to match different communication systems with the goal of approaching the theoretical limits of these systems. The first case we study is the three-node relay channel with the decode-andforward (DF) protocol. According to the structure of the relay channel, we propose bilayer expurgated and bilayer lengthened LDPC convolutional codes. The constructions are realized by adding new check nodes and new variable nodes, respectively, to the single-layer codes. Both the code constructions are proved to be optimal in the sense that the highest possible transmission rate of the relay channel with the DF protocol is achieved. The bilayer codes have two types of edges. We generalize the two-edge-type expurgated structure to the multi-edge-type (MET) construction by carrying out the expurgation multiple times. The resulting MET expurgated LDPC convolutional codes are well suited for the broadcast channel with receiver side information. We prove that the capacity region of the bidirectional broadcast channel with common message is achieved by using the proposed code. The application can be extended to the multi-user case, and a similar conclusion can be obtained. Both the expurgated and lengthened constructions are based on graph extensions in one dimension, i.e., either new check nodes or new variable nodes are added to the existing structure. When carrying out the graph extension properly in two dimensions, we obtain the rate-compatible (RC) LDPC convolutional codes. The proposed RC LDPC convolutional code family theoretically covers all the rational rates from 0 to 1.We prove analytically that all the members of the RC family are simultaneously capacity-achieving. The RC LDPC convolutional code family enables a system to conveniently adapt to varying channel conditions. We then propose applications of the RC family in the case of hybrid automatic repeat-request and for dynamic decode-and-forward relaying. Finally, we generalize the MET LDPC convolutional code construction and provide general guidance regarding parameter selection, aiming for a capacity-achieving performance. We show that, if a choice of degree distribution makes the MET LDPC block code an overall regular code, the same choice will improve the belief-propagation threshold of its MET convolutional counterpart to the maximum a posteriori threshold. If we then increase the node degrees while keeping the rate fixed, a capacity-achieving MET LDPC convolutional code is obtained. With this generalized theory, the design of good distributed codes for multi-terminal networks is simplified to a problem of finding the proper relationship of the node degrees.QC 20121213</p