1. Time Varying Convolutional Codes for Punctured Turbocodes
Yannick Saouter
Claude Berrou
09/04/2019

2. Turbocodes
Turbo code encoding : data frame X=(x1,…,xn) is permuted by  to give X’=(x(1),…,x(n)). The two frames are encoded by a RSC and gives redundancies Y1 and Y2.
Turbo code decoding : Iterative MAP or SUB-MAP.
Invented by Claude Berrou and Alain Glavieux [BerrouGlavieux93]

3. Criteria for good performance
Low autocorrelations in permutations : Lee distance (S-Random) [DolinarDivsalar95], autocorrelation girth [Saouter10] …
Large minimum distance for turbocodes as error-correcting codes.
Theoretical result: turbocode with N information symbols, minimum distance D, RSC components with minimum distance d [Breiling04]
Good turbocodes requires large d for components.
DK d log(N)

4. Heller’s bound
Heller’s bound : Convolutional codes with encoding rate R=1/n, constraint length  and minimum distance d [Heller68]:
Example : R=1/2
dmin1k (n/2).2k/(2k-1).(+k-1)
Heller’s bound vs. Non periodic convolutional codes
Bound unreached for =5, 8, 10, 12, 14

5. Periodic convolutional codes eventually meet Heller’s bound
The period 4 convolutional code ([37,27], [37,35], [33,25], [33,25]) has d=8 for =5 [Lee89].
In most cases, Heller’s bound can be reached

6. From punctured non systematic convolutional codes …
Search for non systematic high rate punctured convolutional codes [Yasuda et al. 84].
Criterion PV : bit error probability with Viterbi decoding algorithm.
Let R=n/(n+1) and C optimal 1/2 convolutional code.
Every n periods, 2n bits are produced by C.
Puncture at each period one of the two redundancies produced.
Final rate = n/(n+1).
Compute PV for 2n-1 possibilities and select the minimum value.
Example : for C=(15,17) and n=3, best puncturing pattern (110,101).

7. … to punctured recursive systematic codes
Previous codes naturally produce PRS when rendered recursive.
Mother code C=(R1,R2), puncturing pattern (1b…b,1b…b) of length n, if b=0 (resp. 1) in the first pattern then b=1 (resp. 0) in the second.
If pattern index >1, no choice possible.
If pattern=(0,1) then PRS=R1/R2.
If pattern=(1,0) then PRS=R2/R1.
At index pattern 0, choose either R1/R2 or R2/R1.
Example : C=(15,17), n=3, pattern=(110,101). PRS=(17/15,17/15,15/17) or (15/17,17/15,15/17).

8. New criteria for search
Decoding of convolutional codes: nowadays Maximum a Posteriori or SUBMAP.
MAP minimizes bit error probability.
Upper bound: rate R=k/n, minimum distance d, multiplicity ad [Viterbi71]
Tail-biting codes : encoding initial state = encoding final state.
Maximize d. If equality, minimize ad.
Codes are checked for circularity.

9. Mother codes : (13,15) for = 4 and (23,35) for  =5.
Summary of the search
Mother codes : (13,15) for = 4 and (23,35) for  =5.
Common choices in turbocode domain.
Best PRS for mother code (13,15)
Best PRS for mother code (23,35)

10. Application to turbocodes
Simulations : Time varying vs. Standard convolutional codes
8 iterations of Max-Log-MAP
S-Random Permutation, tail biting
8-state binary, frame length = 720 bits
16-state binary, frame length = 560 bits

11. Decoding : done iteratively through state lattice
Decoding complexity
Decoding : done iteratively through state lattice
Now, 2 state lattices instead of 1.
Small difference : switch of two branch metrics.
Can be done easily  small increase of VLSI complexity

12. Further axes of research
Time Varying convolutional codes can improve turbocode performance
S-Random permutations are not the best (QPP, ARP, DRP)
Design specific permutations for TV turbocodes.
Drawback of TV turbocodes : one encoder for each encoding rate.
Obtain improvements with punctured full rate TV convolutional code.

13. Thank you for your attention !