Turbocodes

Turbo codes are a class of high-performance error-correcting codes introduced in 1993 by Claude Berrou, Alain Glavieux, and P. Thitimajshima. They approach the Shannon limit on noisy channels by combining two recursive systematic convolutional encoders in parallel, separated by an interleaver, and decoding iteratively.

Structure: An information sequence is encoded by two recursive systematic convolutional (RSC) encoders in parallel, with an interleaver between them. The two parity streams, together with the original data, form the codeword; most implementations are rate 1/3, adjustable by puncturing.

Decoding: Two soft-output decoders corresponding to the constituent encoders exchange extrinsic information iteratively, typically using the BCJR (MAP) algorithm or its approximations. The process embodies the turbo principle of refining estimates by passing information back and forth.

Performance and limits: Turbo codes closely approach the Shannon limit for long block lengths, with strong error correction at moderate SNRs. Short blocks suffer higher error floors; decoding complexity and latency grow with the number of iterations and the interleaver size.

Applications and impact: Since their introduction, turbo codes have influenced many mobile and broadcast systems and spurred extensive research on iterative decoding and related codes, including serial and parallel concatenated variants.

Variants: The standard form uses parallel concatenation of two RSC encoders; variants include serial schemes and higher-rate punctured versions.