About rate-1 codes as inner codes 5th International Symposium on Turbo Codes & Related Topics About rate-1 codes as inner codes C. Berrou, A. Graell i Amat, Y. Ould Cheikh Mouhamedou September 2008 Aujourd'hui, contrairement au passé, on peut imaginer des milliers de codes dès lors que le décodage itératif est possible (composants SISO disponibles)

Rate-1 codes are used, in conjunction with permutation, to increase the minimum Hamming distance of a coding scheme Example:

Permutation to devise with great care! Regular permutation

acts as an R = ½ decoder with dmin= 3 Only conceivable in an iterative process, with systematic loss on convergence threshold (multiplication of errors at the first iteration) acts as an R = ½ decoder with dmin= 3

Question: is there another encoder with larger dmin (for R = 1/2) and no increase in error multiplication (for R = 1)? (and in passing enabling tail-biting termination) For instance, dmin = 4? 3 errors (at least) 1 error input output The answer is obviously: no...

As a rate-1 encoder, let us consider for instance As polynomials 15 and 13 are relatively prime, the input cannot be infered from the output There are 7 candidate polynomials for parity, different from the recursivity polynomial and including the first tap

Encoding with time-varying parity construction (8-state) The multiplication of error is equal to 2 and dmin = 4! But the code is not efficient (large multiplicity)

Encoding with time-varying parity construction (4-state) In a non-systematic version, this would have been named a catastrophic code

Breaking the "catastrophic" nature of the code For each replacement, 3 input values cannot be infered from parity L w2 w2 Which value for L?

The choice of L L = 30 10% not decoded at the first iteration

Exit charts

Possible applications Accumulate-Repeat-Accumulate codes (A. Abbasfar, D. Divsalar, and K. Yao IEEE Trans. Commun., April 2007) 3D-turbo codes ("Adding a Rate-1 Third Dimension to Turbo Codes", C. Berrou, A. Graell i Amat, Y. Ould Cheikh Mouhamedou, C. Douillard, Y. Saouter, ITW 2007)

3D-turbo codes (with double-binary component codes) Max-Log-MAP

Going back over 8-state replaced by w5 every L replaced by w5 For each replacement, 4 input values cannot be infered from parity

8-state versus 4-state 8-state, L=21 19% not decoded at the first iteration 4-state , L=10 30% not decoded at the first iteration R = 1

Conclusions In the context of iterative decoding, rate-1 codes are powerful components to construct concatenated codes and/or to increase minimum Hamming distances of existing schemes There are possible choices other than the classical 2-state accumulator code, with better performance Generally speaking, time-varying construction offer interesting perspectives in the search for powerful codes