We consider the design and analysis of the efficiently-encodable rate-compatible ($E^2RC$) irregular LDPC codes proposed in previous work. In this work we introduce semi-structured $E^2RC$-like codes and protograph $E^2RC$ codes. EXIT chart based methods are developed for the design of semi-structured $E^2RC$-like codes that allow us to determine near-optimal degree distributions for the systematic part of the code while taking into account the structure of the deterministic parity part, thus resolving one of the open issues in the original construction. We develop a fast EXIT function computation method that does not rely on Monte-Carlo simulations and can be used in other scenarios as well. Our approach allows us to jointly optimize code performance across the range of rates under puncturing. We then consider protograph $E^2RC$ codes (that have a protograph representation) and propose rules for designing a family of rate-compatible punctured protographs with low thresholds. For both the semi-structured and protograph $E^2RC$ families we obtain codes whose gap to capacity is at most 0.3 dB across the range of rates when the maximum variable node degree is twenty.
Low-density parity-check (LDPC) codes [1] have found widespread acceptance in different areas due to their superior performance and low complexity decoding. In this paper, we investigate rate-compatible punctured LDPC codes that have the flexibility of operating at different code rates while having a single encoder-decoder pair. Rate-compatible punctured codes are defined by specifying a systematic mother code that operates at the lowest code rate. The parity bits of higher rate codes in a rate-compatible code family are subsets of the parity bits of lower rate codes. A number of papers have investigated issues around the design of good rate-compatible punctured LDPC codes. The work of [2] presents methods for finding optimal degree distributions for puncturing. In [3] [4] [5], algorithms for finding good puncturing patterns for a given mother code were proposed. There have also been attempts to design mother codes (along with puncturing patterns) with good performance under puncturing [6][7] [8].
E 2 RC codes introduced in [6] are linear-time encodable and have good puncturing performance across a wide range of code rates. In this work we present systematic approaches for the design and analysis of E 2 RC-like codes. Let H = [H 1 |H 2 ] denote the parity check matrix of a systematic LDPC code where H 1 denotes the systematic part and H 2 the parity part. We address the design of two types of codes in our work as explained below.
i) Semi-structured E 2 RC-like codes. In these codes the parity part H 2 is deterministic. We use the lower triangular form introduced in [6] and introduce a protograph structure for the H 2 part. An example is shown in Fig. 1.
We assume a random edge interleaver between systematic variable nodes and check nodes, which divides the code into a structured part and an unstructured part, as shown in Fig. 1. We solve the problem of finding optimal degree distributions for the unstructured part in this case for optimizing the rate-compatible codes at any specified punctured code rate(s). ii) Structured E 2 RC-like codes. These codes are protograph codes as introduced in [9]. The distinguishing feature is that the parity part of the protograph has an E 2 RC structure. We demonstrate that very good rate-compatible punctured code families can be obtained using the design rules we propose for the protograph construction. The protograph structure is especially valuable in practical applications as it allows parallelized decoding and requires significantly less storage space for the description of the parity-check matrix than unstructured codes when circulant permutations are used.
We obtain semi-structured E 2 RC codes that have a small gap to capacity across the range of puncturing rates. Furthermore, we present optimized quasi-cyclic protograph codes based on the E 2 RC structure and demonstrate that very good performance can be obtained with them. This paper is organized as follows. In Section II, we briefly discuss the main contributions of our work. Section IV presents our new method for the design of semi-structured E 2 RC codes. We also discuss the method of predicting the puncturing performance of semi-structured E 2 RC codes and the joint optimization of our codes at any specified punctured code rates. We explain the construction of protograph E 2 RC codes in Section V, and Section VI outlines our conclusions.
We first outline the issues left unresolved in the work of [6].
a) The original construction of E 2 RC codes proposed the special H 2 (parity part) structure of the parity-check matrix H. However the design of appropriate degree sequences for the H 1 (information part) based on the constrained H 2 structure, was not discussed. In [6], the authors used degree sequences designed for standard irregular codes and constructed H to match these distributions as closely as possible. b) The construction technique did not provide any means of optimizing code performance at any particular puncturing rate or across all rates simultaneously. c) As pointed out by an anonymous reviewer, the original E 2 RC codes suffer from high error floors [10] at the mother code rate. As shown in [10], this is because the H 2 structure causes the maximum check node degree to be large. d) The original E 2 RC codes work with completely random interleavers, that are hard to implement in practice.
In this paper, we resolve each of the issues discussed above. We briefly overview the main contributions below. i) Systematic design techniques for E 2 RC-like codes. Note that the analysis of E 2 RC codes does not follow directly from the analysis of related codes such as systematic IRA codes [11][12]. This is because the structured part of IRA codes is symmetric while that of E 2 RC codes is quite asymmetric. In [12], four methods were proposed for the design of IRA codes. The first two methods implicitly assumed one edge type in the accumulator part which was justified by the symmetry of the part. Together
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