High-Rate Spatially Coupled LDPC Codes Based on Massey's Convolutional Self-Orthogonal Codes

TL;DR

This work introduces high-rate SC-LDPC codes constructed from rate $R=(n-1)/n$ convolutional self-orthogonal codes (CSOCs) by interpreting their parity-check graphs as convolutional protographs and lifting them with permutation matrices for BP-based sliding-window decoding. To address degree-1 variable nodes in systematic CSOC protographs, a non-systematic variant is proposed that preserves girth at least six and minimum free distance $d_{free}=J+1$, while allowing systematic encoding. Numerical results show excellent high-rate performance and competitive thresholds, with iterative decoding thresholds approaching the MAP limit as lifting factor $M$ and termination length $L$ grow; comparisons indicate non-systematic CSOC-based SC-LDPC codes can outperform systematic counterparts. The proposed CSOC-based construction offers flexible, low-latency high-rate SC-LDPC codes with practical lifting options (e.g., circulant liftings) and solid distance properties, making them attractive for high-throughput communication systems.

Abstract

In this paper, we study a new class of high-rate spatially coupled LDPC (SC-LDPC) codes based on the convolutional self-orthogonal codes (CSOCs) first introduced by Massey. The SC-LDPC codes are constructed by treating the irregular graph corresponding to the parity-check matrix of a systematic rate R = (n - 1)/n CSOC as a convolutional protograph. The protograph can then be lifted using permutation matrices to generate a high-rate SC-LDPC code whose strength depends on the lifting factor. The SC-LDPC codes constructed in this fashion can be decoded using iterative belief propagation (BP) based sliding window decoding (SWD). A non-systematic version of a CSOC parity-check matrix is then proposed by making a slight modification to the systematic construction. The non-systematic parity-check matrix corresponds to a regular protograph whose degree profile depends on the rate and error-correcting capability of the underlying CSOC. Even though the parity-check matrix is in non-systematic form, we show how systematic encoding can still be performed. We also show that the non-systematic convolutional protograph has a guaranteed girth and free distance and that these properties carry over to the lifted versions. Finally, numerical results are included demonstrating that CSOC-based SC-LDPC codes (i) achieve excellent performance at very high rates, (ii) have performance at least as good as that of SC-LDPC codes constructed from convolutional protographs commonly found in the literature, and (iii) have iterative decoding thresholds comparable to those of existing SC-LDPC code designs.

Figures (6)

• Figure 1: The (a) unterminated and (b) terminated convolutional protograph corresponding to the parity-check matrix $\bf{H}_{\rm{ns}}$ of \ref{['eq:Hns_m13J4']}.
• Figure 2: Block diagram of a SWD with window size $W$ constraint lengths for an SC-LDPC code.
• Figure 3: Performance comparison of two CSOCs, one systematic and one non-systematic, with $R=2/3$, $J=4$, and $L = 200$.
• Figure 4: BER performance of two high-rate CSOC-based SC-LDPC codes.
• Figure 5: (a) The multi-edge protograph representing the base matrix $\bf{B}$, (b) The corresponding single-edge convolutional protograph obtained by classical edge spreading.

Theorems & Definitions (2)

• Theorem 1
• proof