Two classes of turbo codes constructed on high-order finite fields are introduced. The codes are derived from a particular protograph sub-ensemble of the (2,3) regular low-density parity-check (LDPC) code ensemble. The first construction results in a parallel concatenation of two non-binary, time-variant accumulators. The second construction consists of the serial concatenation of a non-binary time-variant differentiator with a non-binary time-variant accumulator, and provides a highly structured flexible encoding scheme for (2,4) LDPC codes. A cycle graph representation is also provided. The proposed codes can be decoded efficiently either as LDPC codes (via belief propagation decoding on their bipartite graphs) or as turbo codes (via the forward-backward algorithm applied to the component code trellises) by means of the fast Fourier transform. The proposed codes provide remarkable coding gains (more than 1 dB at a codeword error rate 10-4) over binary LDPC and turbo codes in the moderate-short block length regime.

Short Turbo Codes over High Order Fields

Two classes of turbo codes over high-order finite fields are introduced. The codes are derived from a particular protograph sub-ensemble of the (dv=2,dc=3) low-density parity-check code ensemble. A first construction is derived as a parallel concatenation of two non-binary, time-variant accumulators. The second construction is based on the serial concatenation of a non-binary, time-variant differentiator and of a non-binary, time-variant accumulator, and provides a highly-structured flexible encoding scheme for (dv=2,dc=4) ensemble codes. A cycle graph representation is provided. The proposed codes can be decoded efficiently either as low-density parity-check codes (via belief propagation decoding over the codes bipartite graph) or as turbo codes (via the forward-backward algorithm applied to the component codes trellis). The forward-backward algorithm for symbol maximum a posteriori decoding of the component codes is illustrated and simplified by means of the fast Fourier transform. The proposed codes provide remarkable gains (~ 1 dB) over binary low-density parity-check and turbo codes in the moderate-short block regimes.

Turbo Codes Based on Time-Variant Memory-1 Convolutional Codes over Fq

Two classes of turbo codes over high-order finite fields are introduced The codes are derived from a particular protograph sub-ensemble of the (dv=2,dc=3) low-density parity-check code ensemble A first construction is derived as a parallel concatenation of two non-binary, time-variant accumulators The second construction is based on the serial concatenation of a non-binary, time-variant differentiator and of a non-binary, time-variant accumulator, and provides a highly-structured flexible encoding scheme for (dv=2,dc=4 ensemble codes A cycle graph representation is provided The proposed codes can be decoded efficiently either as low-density parity-check codes (via belief propagation decoding over the codes bipartite graph) or as turbo codes (via the forward-backward algorithm applied to the component codes trellis) The forward-backward algorithm for symbol maximum a posteriori decoding of the component codes is developed and simplified by means of the fast Fourier transform The proposed codes provide remarkable gains (~1 dB) over binary low-density parity-check and turbo codes in the moderate-short block regimes

The iterative decoding structure and component maximum a posteriori decoders used for decoding binary concatenated codes can be extended to the nonbinary domain. This paper considers turbo codes over nonbinary rings, specifically ternary, quaternary, penternary, hexernary, and octernary codes. The best rate-1/2 component codes are determined using a practical search algorithm. The performance of the resulting rate-1/3 turbo codes on an additive white Gaussian noise channel using q-ary phase-shift keying modulation is given.

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Rate-1/2 component codes for nonbinary turbo codes

In this paper we consider the use of q-ary turbo codes on abstract q-ary channels. Simulations show that our 16-ary codes perform well on the q-ary symmetric channel. This validates their suitability as outer codes in non-binary applications. We also compare with the performance of same codes on conventional PSK and QAM modulation, and demonstrate an application of these codes to the insertion/deletion channel.

Non-binary turbo codes and applications

This paper describes a construction technique for q-ary turbo codes that computes good recursive systematic convolutional q-ary constituent codes with constraint length v/spl les/5 for q=2/sup m/, m=2, 3, and 4. The construction technique, based on the algorithm of Benedetto, Garello and Montorsi (see IEEE Trans. Commun., vol.46, p.1101-1105, 1998), determines the codes with maximum d/sub i/ for i=2, 3, and 4 and minimum codeword multiplicity, where d/sub i/ is the minimum weight of all code sequences with input weight i. Due to the large number of encoder states involved, standard weight distribution calculations are difficult. The construction algorithm employed is a computer search that generates all possible terminating sequences of weight 2, 3, and 4 to use as inputs to the set of allowable encoders.

G.S. White

Papers

Construction and performance of q-ary turbo codes for use with M-ary modulation techniques