Analysis and Design of Serially Concatenated LDGM Codes

TL;DR

With an efficient DDE-based optimization approach that utilizes the necessary condition for the successful decoding of these codes, optimized SCLDGM codes that approach the Shannon limit are constructed.

Abstract

In this paper, we first present the asymptotic performance of serially concatenated low-density generator-matrix (SCLDGM) codes for binary input additive white Gaussian noise channels using discretized density evolution (DDE). We then provide a necessary condition for the successful decoding of these codes. The error-floor analysis along with the lower bound formulas for both LDGM and SCLDGM codes are also provided and verified. We further show that by concatenating inner LDGM codes with a high-rate outer LDPC code instead of concatenating two LDGM codes as in SCLDGM codes, good codes without error floors can be constructed. Finally, with an efficient DDE-based optimization approach that utilizes the necessary condition for the successful decoding, we construct optimized SCLDGM codes that approach the Shannon limit. The improved performance of our optimized SCLDGM codes is demonstrated through both asymptotic and simulation results.

Figures (15)

• Figure 1: Concatenated Codes
• Figure 2: An example of bi-partite graphs representing inner and outer decoders.
• Figure 3: Two-step vs. joint decoding performance of a SCLDGM code. The inner code used is half-rate $(7,7)$-LDGM code. The outer code used is $50/51$-rate $(4,200)$-LDGM code. Overall code-rate = 0.49.
• Figure 4: Asymptotic performance of $\left(d_v,d_c\right)$- LDGM codes, code-rate = 50/51.
• Figure 5: Asymptotic performance of various SCLDGM codes. Overall code-rate = 0.49.