Construction of quasi-cyclic LDPC codes for AWGN and binary erasure channels: a finite field approach

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. Index Terms--Burst, dispersion, erasure, error floor, low-density parity-check (LDPC) codes, masking, quasi-cyclic (QC) codes, zero span.