 
Summary: 1812 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004
Simple MAP Decoding of FirstOrder ReedMuller and
Hamming Codes
Alexei Ashikhmin, Member, IEEE, and
Simon Litsyn, Senior Member, IEEE
AbstractA maximum a posteriori (MAP) probability decoder of a block
code minimizes the probability of error for each transmitted symbol sepa
rately. The standard way of implementing MAP decoding of a linear code
is the BahlCockeJelinekRaviv (BCJR) algorithm, which is based on a
trellis representation of the code. The complexity of the BCJR algorithm
for the firstorder ReedMuller (RM1) codes and Hamming codes is pro
portional to , where is the code's length. In this correspondence, we
present new MAP decoding algorithms for binary and nonbinary RM1
and Hamming codes. The proposed algorithms have complexities propor
tional to log , where is the alphabet size. In particular, for the
binary codes this yields complexity of order log .
Index TermsHamming codes, maximum a posteriori (MAP) decoding,
ReedMuller codes.
I. INTRODUCTION
The firstorder ReedMuller (RM1) codes is a family of lowrate
