Summary: 1812 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 8, AUGUST 2004
Simple MAP Decoding of First-Order Reed­Muller and
Hamming Codes
Alexei Ashikhmin, Member, IEEE, and
Simon Litsyn, Senior Member, IEEE
Abstract--A 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 first-order Reed­Muller (RM-1) 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 RM-1
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 Terms--Hamming codes, maximum a posteriori (MAP) decoding,
Reed­Muller codes.
I. INTRODUCTION
The first-order Reed­Muller (RM-1) codes is a family of low-rate

Source: Ashikhmin, Alexei - Enabling Computing Technologies, Bell Laboratories

Collections: Engineering; Mathematics; Computer Technologies and Information Sciences