Low-rate turbo-Hadamard codes

This paper is concerned with a class of low-rate codes constructed from Hadamard code arrays. A recursive encoding principle is employed to introduce an interleaving gain. Very simple trellis codes with only two or four states are sufficient for this purpose, and the decoding cost involved in the trellis part is typically negligible. Both simulation and analytical results are provided to demonstrate the advantages of the proposed scheme. The proposed scheme is of theoretical interest as it can achieve performance of BER=10^-5 at E_b/N₀≈-1.2dB (only about 0.4 dB away from the ultimate low-rate Shannon limit) with an information block size of 65534. To the authors´ knowledge, this is the best result achieved to date with respect to the ultimate Shannon limit. With regard to practical issues, the decoding complexity of the proposed code is considerably lower than that of existing low-rate turbo-type codes with comparable performance.