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.