capacity

transmitted over a communication channel with an error probability

that will vanish if we allow the blocklength to get infinitely

large. While this result is of tremendous theoretical importance, the

reality of practical systems looks quite different: no communication

system will tolerate an infinite delay caused by an extremely large

blocklength, nor can it deal with the computational complexity of

decoding such huge codewords. On the other hand, it is not necessary

to have an error probability that is exactly zero either, a small, but

finite value will suffice.

Therefore, the question arises what can be done in a practical

scheme. In particular, what is the maximal rate at which information

can be transmitted over a communication channel for a given fixed

maximum blocklength (i.e., a fixed maximum delay) if we allow a

certain maximal probability of error? In this project, we have started

to study these questions.

Block-codes with very short blocklength over the most general binary

hannel, the

investigated. It is shown that for only two possible messages,

flip-flop codes are optimal, however, depending on the blocklength and

the channel parameters, not necessarily the linear flip-flop

code. Further it is shown that the optimal decoding rule is a

threshold rule. Some fundamental dependencies of the best code on the

channel are given.

Block-codes with a very small number of codewords are investigated for

the two special binary memoryless channels, the

channel (BSC)

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

are optimal for any

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

two and four messages, the optimal code is shown to be linear. For the

ZC a 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.

finite blocklengths, ML, optimal codes, Z-channel.

-||- _|_ _|_ / __|__ 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: Tue Jan 31 17:24:31 UTC+8 2012