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Turbo coding is an iterated soft-decoding scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the Shannon limit.
A code has all-symbol locality and availability if every code symbol can be recovered from disjoint repair sets of other symbols, each set of size at most symbols. Such codes are called ( r , t ) a {\displaystyle (r,t)_{a}} -LRC.
To convolutionally encode data, start with k memory registers, each holding one input bit.Unless otherwise specified, all memory registers start with a value of 0. The encoder has n modulo-2 adders (a modulo 2 adder can be implemented with a single Boolean XOR gate, where the logic is: 0+0 = 0, 0+1 = 1, 1+0 = 1, 1+1 = 0), and n generator polynomials — one for each adder (see figure below).
In this example, we shall encode 14 bits of message with a 3-bit CRC, with a polynomial x 3 + x + 1. The polynomial is written in binary as the coefficients; a 3rd-degree polynomial has 4 coefficients (1x 3 + 0x 2 + 1x + 1). In this case, the coefficients are 1, 0, 1 and 1. The result of the calculation is 3 bits long, which is why it is called ...
Given a prime number q and prime power q m with positive integers m and d such that d ≤ q m − 1, a primitive narrow-sense BCH code over the finite field (or Galois field) GF(q) with code length n = q m − 1 and distance at least d is constructed by the following method.
Turbo codes, as described first in 1993, implemented a parallel concatenation of two convolutional codes, with an interleaver between the two codes and an iterative decoder that passes information forth and back between the codes. [6] This design has a better performance than any previously conceived concatenated codes.
The on-line textbook: Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay, contains chapters on elementary error-correcting codes; on the theoretical limits of error-correction; and on the latest state-of-the-art error-correcting codes, including low-density parity-check codes, turbo codes, and fountain codes.
Fig 1 is an example of a SCCC. Fig. 1. SCCC Encoder. The example encoder is composed of a 16-state outer convolutional code and a 2-state inner convolutional code linked by an interleaver. The natural code rate of the configuration shown is 1/4, however, the inner and/or outer codes may be punctured to achieve higher code rates as needed.