Low rate turbo-Hadamard codes

Abstract

Abstract-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 0 1.2 dB (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.Index Terms-A posteriori probability (APP) decoding, Hadamard codes, iterative decoding, spread spectrum, turbo codes.

Interleave‐division multiple access (IDMA) using low‐rate layered LDGM codes

“…Regarding complexity, Table II shows the number of operations of a rate 1/7.7 layered LDGM scheme and the Turbo‐Hadamard code with the same rate from Reference 7. Although the number of exponential operations in the proposed scheme is larger than that of those based on Turbo‐Hadamard codes, there exist approximations for the decoding of parity checks that provide substantial reductions in complexity while keeping practically the same performance 22.…”

“…It is natural then to substitute the concatenation of code plus spreading by a single low‐rate code. In Reference 5, the use of Turbo‐Hadamard 7 codes is proposed, showing that the obtained performance is better than that of a conventional medium‐rate Turbo code concatenated with a spreading operation.…”

Interleave-division-multiplexing space-time codes

“…In this case, R, I 118 and the low-rate turbo-Hadamard code [4] can be used. A turbo-Hadamard code with M constituent codes has an overall rate of r / ( r + M ~( 2 ' -r ) ) [4], where r is the order of the Hadamard code used. We transmit the information sequence, Rayleigh fading channels.…”

“…Thus, a low-rate code can be used as C to maximize the coding gain. A good choice is the turbo-Hadamard code studied in [4] that can achieve performance close to the ultimate Shannon limit of -1.6dB in AWGN.…”

Scrambled Coded Multiple Access

“…This approach has been successfully implemented for traditional multiple access schemes such as TDMA, FDMA or CDMA. In fact, similar near optimum FEC codes such as turbo-Hadamard codes [3] or low-rate turbo codes [4] have been used in the context of IDMA as well. On the other hand more recently, it has been shown that the proper choice of FEC for IDMA or SCMA should be a function of the number of users or the aggregate throughput [5].…”