the two special binary memoryless channels, the binary symmetric

channel (BSC) and the Z-channel (ZC). The optimal (in the sense of

minimum average error probability, using maximum likelihood decoding)

code structure is derived for the cases of two, three, and four

codewords and an arbitrary blocklength. It is shown that for two

possible messages, on a BSC, the so-called

t

is optimal. For codes with three or four messages it is shown that the

so-called

the type depends on the blocklength. For all cases an algorithm is

presented that constructs an optimal code for blocklength

recursively from an optimal code of length

recursive optimal code design is conjectured in the case of five

possible messages.

The derivation of these optimal codes relies heavily on a new approach

of constructing and analyzing the code-matrix not row-wise

(codewords), but

prove that the minimum Hamming distance might be the wrong design

criterion for optimal codes even for very symmetric channels like the

BSC.

-||- _|_ _|_ / __|__ Stefan M. Moser

[-] --__|__ /__\ /__ Senior Scientist, ETH Zurich, Switzerland

_|_ -- --|- _ / / Adjunct Professor, National Yang Ming Chiao Tung University, Taiwan

/ \ [] \| |_| / \/ Web: https://moser-isi.ethz.ch/

Last modified: Mon Nov 14 13:11:04 UTC+8 2011