Optimal block-codes with a very small number of codewords are
investigated for the binary input discrete memoryless channels. Those
channels are the binary asymmetric channel (BAC), including the two
special cases of the binary symmetric channel (BSC) and the Z-channel
(ZC). The binary erasure channel (BEC) is a common used channel with
ternary output. For the asymmetric channels, a general BAC, it is
shown that so-called flip codes are optimal codes with two codewords.
The optimal (in the sense of minimum average error probability, using
maximum likelihood decoding) code structure is derived for the ZC in
the cases of two, three, and four codewords and an arbitrary finite
blocklength. For the symmetric channels, the BSC and the BEC, the
optimal code structure is derived with at most three codewords and an
arbitrary finite blocklength, a statement for linear optimal codes
with four codes is also given.

The derivation of these optimal codes relies heavily on a new approach
of constructing and analyzing the codebook matrix not row-wise
(codewords), but column-wise. This new tool allows an elegant
definition of interesting code families that is recursive in the
blocklength n and admits their exact analysis of error performance
that is not based on the union bound or other approximations.

-||-   _|_ _|_     /    __|__   Stefan M. Moser
[-]     --__|__   /__\    /__   Senior Researcher & Lecturer, ETH Zurich, Switzerland
_|_     -- --|-    _     /  /   Adj. Professor, National Chiao Tung University (NCTU), Taiwan
/ \     []  \|    |_|   / \/    Web: http://moser-isi.ethz.ch/

Last modified: Tue Aug 20 06:28:40 UTC+8 2013