On the Design of Arbitrarily Low-Rate Turbo-Codes

This paper presents a design criteria for arbitrarily low-rate parallel concatenated convolutional codes (PCCCs). The purpose of this work is to find a family of turbo codes that work as close to the ultimate low-rate Shannon limit E_b/N_o sime -1.59 dB as possible, given a certain constraint in the number of states of the constituent trellis codes and in the interleaver-length. We propose an optimization criteria and reduce the turbo-design problem to the design of block codes for the assignment of output sequences to the trellis branches of the constituent encoders. We show that BCH codes concatenated with repetition codes are optimal for labeling. Moreover, we show that for a fixed number of trellis states these codes achieve arbitrarily low rates, and hence arbitrarily low SNRs, with practically the same performance in terms of E_b/N_o. Simulation results are shown for 8-state and 16-state turbo codes with rates as low as 1/505, which with an interleaver-length of 8192 provide a BER sime 10^-5 at an SNR sime -27.6 dB (E_b/N_o sime -0.55 dB), around 1 dB away from the ultimate low-rate Shannon limit.