A rate-compatible family of protograph-based LDPC codes built by expurgation and lengthening

We construct a protograph-based rate-compatible family of low-density parity-check (LDPC) codes that cover a very wide range of rates from 1/2 to 16/17, perform within about 0.5 dB of their capacity limits for all rates, and can be decoded conveniently and efficiently with a common hardware implementation. In contrast to alternative methods that create codes of different rates by puncturing, shortening, or expurgation, our method uses a combination of expurgation and lengthening (equivalent to an unusual extension) to produce lower-rate codes. Advantages compared to the alternative methods include roughly uniform utilization of common family decoder hardware for different rates, implementation with uniformly low maximum check node degrees despite high maximum rate, and a large fixed portion of the protograph that can be labeled with a fixed set of edge permutations for all rates. We apply this method to create a rate-compatible code family anchored by a particular code of (nominal) rate 7/8 and length n = 8176 designed by Kou, Lin and Fossorier [1] whose edge permutations are determined by Euclidean geometries (EG). All members of our family retain all of the EG-designed edges and circulant permutations of this anchor code, and this helps to avoid weak spots in the code graph that usually arise when edge permutations are assigned by greedy algorithms. There are also varying numbers of auxiliary and ancillary checks, variables, and edges, allowing realization of different rates, with fixed (nominal) dimension k = 8176 for all rates. Simulations show that all members of this family achieve steeply falling error rate curves without detectable error floors, at least to codeword error rates of about 10^-6