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

Abstract

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over binary low-density parity-check and turbo codes in the moderate-short block regimes.Comment: This paper has been withdrawn by the authors. To be presented at IEEE ICC 2011, Kyoto, Japan. June 201

Short low-rate non-binary turbo codes

“…This feature is even more appealing, considering that the forward-backward algorithm over the component trellises of the turbo codes of [11] can be performed efficiently with complexity O(q log q). This is again due to the order-q FHT which is used to dualize the check node message passing rule [11]. Thus, when orderq Hadamard codes or first order length-q RM codes are employed as outer codes, the overall decoding complexity is O(q log q).…”

“…In fact, when a Hadamard code tailored to a field order q is used, the overall coding rate is R = (1/3) × (log 2 q/q), whereas, if a first order RM code is adopted, R = (1/3)× (2 log 2 q/q). Thus, the use of large field orders, which leads to turbo codes with excellent performance [11], turns in extremely-low coding rates. As an example, if q = 2 8 , the scheme based on an inner Hadamard code has a coding rate R = 1/96, which is doubled if an inner first order RM code is used.…”

“…The second set of parity symbols p (2) is obtained in a similar way after permuting the symbols of u according to the interleaving rule i → π(i) (for details see [11]). The codeword is given by w = [u|p (1) |p (2) ].…”

“…The coding rate is given by R O = k O /n O . Due to their excellent performance in the short block length regime, we restrict our analysis to outer codes being non-binary turbo codes based on memory-1 recursive convolutional codes [11], for which iterative decoding can be efficiently implemented thanks to FHTs [14] with complexity O(q log q). The encoder structure is depicted in Figure 1.…”

“…In this paper, a novel low-rate scheme is presented based on the concatenation of inner algebraic codes having good distance properties with the recently-introduced non-binary turbo codes of [11] as outer codes where the inner code dimension matches the turbo code field order q. The proposed concatenation can in principle be also applied to ultra-sparse non-binary LDPC codes.…”