Soft-Decision Stack Decoding of Binary Reed-Muller Codes with Look-Ahead Technique

7th International Workshop on Algebraic and Combinatorial Coding Theory 18-24June2000,Bansko,Bulgaria pp.293-298 Soft-Decision Stack Decoding of Binary Reed-Muller Codes with“Look-Ahead”
Technique
Norbert Stolte,Ulrich Sorger
Institute for Communications Technology
Darmstadt University of Technology(TUD)
e-mail:stolte,uli@nesi.tu-darmstadt.de
Abstract
The performance of a stack algorithm to decode binary Reed-Muller codes is investigated.As metric to evaluate the different elements in the
stack we use a recently derived lower bound on the squared Euclidean
distance.It comprises a lower bound on the metric increase in future
decoding steps,similar to the-algorithm.Simulation results show that
in case of the code at a WER of with reasonable stack
size close to soft-decision ML-decoding performance can be achieved.
1Definitions
We assume BPSK transmission over an additive white Gaussian noise(AWGN) channel.The transmitted codeword denoted by
,,is corrupted by noise and at the receiver the real valued sequence is observed.
It is known[1][2]that the code of length can be composed using the-construction.I.e.with,
and
the obtained generalized concatenated(GC)code is the code.(see Fig.1)Since and are RM codes,they again can be viewed as GC codes and the same construction rule applies.This can be repeated several times and in this way the code is decomposed into
Fig.1:Decomposition of the code into two outer codes simple repetition codes,of different length,were is the dimension of the code.The codeword completely determines the codewords and vice versa.
Soft-decision bounded-minimum-distance(SD-BMD)decoding can be done by decoding only the outer codes and.We define the compo-nents of the soft-input vectors and of the codes and with and as
(1) and
(2) In[3]it was shown that if(1)and(2)are applied successively to calculate the soft-input values of the codes,the maximum-likelihood(ML) decoder decides for the codeword which minimizes the following sum
(3)
For a vector of estimates,, ,the sum
(4)
is a simple lower bound on the least value of(3)in the subcode
which is defined by
following inequality holds
(6) for.In that way similar to the-algorithm [4][5],the increase of in future decoding steps is lower bounded.
2Algorithm and Results
For decoding we use a stack algorithm in which the tree is searched in a breadth-first manner,that is all estimate vectors have same length.In-stead of performing full maximum-likelihood decoding tofind the in(6) a lower bound on is calculated.This is done using a second stack decoder with small stack which employs the metric given in(4). Whenever a stack overflow in the second stack occurs,the respective value of (4)of the element which is not kept in the stack gives a maximum value of
to be used in(6).Hence the metric to order the estimates in the main stack is given by
almost equal to the decoder with parameters.Moreover at a WER of quasi SDML-decoding performance can be achieved if the stack sizes are chosen to be and.
In Fig.4is shown the dependence of the WER on the signal to noise ratio for different stack sizes.At a WER of the performance degradation of the decoders with stack sizes and compared to SDML decoding is about0.4dB and0.2dB,respectively.Moreover compared to the“classical”-decoder the-decoder achieves an additional coding gain of3.4dB at WER=.
References
[1]F.J.MacWilliams,N.J.A.Sloane,”The Theory of Error-Correcting
Codes”,New York,North-Holland,2nd reprint,p.374,1983
[2]G.A.Kabatyansky,”On decoding Reed-Muller codes in semicontinuous
channel”,Proc.Second Int.Workshop Algebr.and Combin.Coding Theory, Leningrad,USSR,pp.87-91,1990
[3]N.Stolte,U.Sorger,G.Sessler,”Sequential Stack Decoding of Binary
Reed-Muller Codes”,ITG Fachbericht,3rd ITG Conference Source and Channel Coding,No.159,pp.63-69,2000
[4]Y.Han, C.Hartmann, C. C.Chen,”Efficient Priority-First Search
Maximum-Likelihood Soft-Decision Decoding of Linear Block Codes”, IEEE rm.Theory,V ol.IT-39,No.5,pp.1514-1523,1993 [5]N.Nilsson,”Principles of Artificial Intelligence”Berlin,Springer Verlag,
1982
Fig.3:Soft-decision decoding WER of the code,decoders with dif-ferent stack sizes
Fig.4:Soft-decision decoding WER of the code,decoders with dif-ferent stack sizes。