Successive Maximization for Systematic Design of Universally Capacity Approaching Rate-Compatible Sequences of LDPC Code Ensembles over Binary-Input Output-Symmetric Memoryless Channels

Abstract

Abstract-A systematic construction of capacity achieving lowdensity parity-check(LDPC) code ensemble sequences over the Binary Erasure Channel (BEC) has been proposed by Saeedi et al. based on a method, here referred to as Successive Maximization (SM). In SM, the fraction of degree-nodes are successively maximized starting from = 2 with the constraint that the ensemble remains convergent over the channel. In this paper, we propose SM to design universally capacity approaching rate-compatible LDPC code ensemble sequences over the general class of Binary-Input Output-Symmetric Memoryless (BIOSM) channels. This is achieved by first generalizing the SM method to other BIOSM channels to design a sequence of capacity approaching ensembles called the parent sequence. The SM principle is then applied to each ensemble within the parent sequence, this time to design rate-compatible puncturing schemes. As part of our results, we extend the stability condition which was previously derived for degree-2 variable nodes to other variable node degrees as well as to the case of ratecompatible codes. Consequently, we prove that using the SM principle, one is able to design universally capacity achieving rate-compatible LDPC code ensemble sequences over the BEC. Unlike the previous results in the literature, the proposed SM approach is naturally extendable to other BIOSM channels. The performance of the rate-compatible schemes designed based on our method is comparable to those designed by optimization.

Randomly Punctured LDPC Codes

“…7 shows numerically calculated values of h G (σ BP (α)) for the C punc 3,6 (50, α) SC-LDPC code ensembles for several puncturing fractions α. Also shown are the predicted thresholds using both (14) and (17). We observe that the prediction is good for small values of α using (17) and good for large values of α using (14).…”

“…This irregular protograph has variable nodes with degrees ranging from 2 to 6 and check nodes with degrees 6 and 7. Puncturing of LDPC-BCs has been investigated extensively in the literature [7]- [14] and good rate-compatible protograph-based codes and code ensembles have been constructed (see, e.g., [5] in the context of graph extending and [17] using puncturing).…”

“…for R ≤ R(α) ≤ 1/θ E . Predicted values of h G (σ BP (α)) obtained using (14) are also included in Fig. 6 as solid lines.…”

“…Coding schemes that can adapt to the changing conditions of time-varying channels while allowing transceivers to employ the same encoder/decoder pair are known as ratecompatible codes [1]. Rate-compatible low-density paritycheck (LDPC) codes have been extensively studied in the literature using code modifying techniques such as information nulling or shortening [2], extending [3]- [6], puncturing [7]- [14], and combining [15].…”

Binary LDPC Codes Decoding Algorithm Based on MRF and FPGA Implementation

“…In fact, the research steps of further enhancing the performance of LDPC codes never stop. LDPC codes have been extensively studied in the literature using code modifying techniques such as information nulling or shortening, extending [7,8], puncturing [9], and combining [10]. We focused on studying the structure of LDPC code and LDPC code decoding algorithm, improved LDPC code decoding algorithm became the most crucial part [11].…”