Linear Rate-Compatible Codes with Degree-1 Extending Variable Nodes Under Iterative Decoding

Abstract

A rate-compatible (RC) code first transmits a set of symbols corresponding to the highest rate. These symbols form the highest-rate code (HRC). If requested by the receiver, the transmitter subsequently sends symbols that lower the rate of the code. Additional symbols are sent until the decoder decodes to a codeword or all symbols of the RC code are exhausted. Consider linear, RC low-density parity-check (LDPC) codes constructed using extending variable nodes of degree 1. That is, every symbol of incremental redundancy (IR) is a linear combination only of symbols of the HRC. We study the convergence of such codes under iterative decoding. We show that the convergence criterion considered after each iteration need only check whether the HRC variable nodes have converged to a codeword. Specifically, there is no need to consider whether the parity checks that generate the IR symbols are satisfied. We substantiate these claims with simulation results of protograph-based raptor-like LDPC (PBRL) codes, which are a family of protograph RC-LDPC codes with the extending structure under consideration. Furthermore, we demonstrate using examples that this extending structure for protograph RC codes is not very far from the optimal extension for a protograph RC code by providing examples of iterative decoding thresholds for PBRL protographs and protographs extended using the optimal degrees for incremental variable nodes.