Construction of Quasi-Cyclic LDPC Codes for AWGN and Binary Erasure Channels: A Finite Field Approach

TLDR

Finite fields were historically employed to build linear block codes, notably cyclic BCH and Reed‑Solomon codes, that achieve large minimum distances for hard‑decision algebraic decoding. The authors present finite‑field construction methods for algebraic low‑density parity‑check (LDPC) codes. The resulting quasi‑cyclic LDPC codes perform exceptionally over AWGN, binary random, and burst erasure channels with iterative decoding, exhibit low error floors, and can be encoded efficiently using shift registers with linear complexity.

Abstract

In the late 1950s and early 1960s, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for hard-decision algebraic decoding, such as Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. This paper shows that finite fields can also be successfully used to construct algebraic low-density parity-check (LDPC) codes for iterative soft-decision decoding. Methods of construction are presented. LDPC codes constructed by these methods are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN), binary random, and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor, and rate of decoding convergence, collectively. Particularly, they have low error floors. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.