Protograph LDPC Codes with Node Degrees at Least 3

Abstract

In this paper we present protograph codes with a small number of degree-3 nodes and one high degree node. The iterative decoding threshold for proposed rate 1/2 codes are lower, by about 0.2 dB, than the best known irregular LDPC codes with degree at least 3. The main motivation is to gain linear minimum distance to achieve low error floor. Also to construct rate-compatible protograph-based LDPC codes for fixed block length that simultaneously achieves low iterative decoding threshold and linear minimum distance. We start with a rate 1/2 protograph LDPC code with degree-3 nodes and one high degree node. Higher rate codes are obtained by connecting check nodes with degree-2 non-transmitted nodes. This is equivalent to constraint combining in the protograph. The condition where all constraints are combined corresponds to the highest rate code. This constraint must be connected to nodes of degree at least three for the graph to have linear minimum distance. Thus having node degree at least 3 for rate 1/2 guarantees linear minimum distance property to be preserved for higher rates. Through examples we show that the iterative decoding threshold as low as 0.544 dB can be achieved for small protographs with node degrees at least three. A family of low- to high-rate codes with minimum distance linearly increasing in block size and with capacity-approaching performance thresholds is presented. FPGA simulation results for a few example codes show that the proposed codes perform as predicted