1. 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)

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

3. Permutation to devise with great care!
Regular permutation

4. 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

5. 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...

6. 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

7. 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)

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

9. 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?

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

11. Exit charts

12. 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)

13. 3D-turbo codes (with double-binary component codes)
Max-Log-MAP

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

15. 8-state versus 4-state
8-state, L= % not decoded at the first iteration
4-state , L= % not decoded at the first iteration
R = 1

16. 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