of nonlinear codes, called

contain many beautiful properties. In particular the subfamily

weak flip codes

Gallager, and Shannon and which was shown to achieve the error

exponent with a fixed number of codewords

generalization of linear codes. The fair weak flip codes are related

to binary nonlinear Hadamard codes.

The exact formula of the average error probability of an arbitrary

code (linear or nonlinear) using maximum likelihood decoding is

provided on binary erasure channels (BECs) with arbitrary erasure

probability

to be a key to the understanding of the codes’ performance. A

generalization of the Plotkin bound for

distance then is purposed for arbitrary binary codes. It is then noted

that the family of

concatenation of several Hadamard linear codes, maximize the minimum

pairwise Hamming distance under certain blocklengths, and the family

of fair weak flip codes can be shown to be the generalized Plotkin

Bound achieving codes.

For the performance analysis of BEC, the globally optimal codes (in

the sense of minimizing the average error probability) for an

arbitrary finite blocklength are given with a number of codewords

satisfying

for the fair codes and weak flip codes. It is also proven that the

fair linear codes is actually strictly suboptimal among all codes for

many values of the blocklength

distance, finite blocklength, weak flip codes, maximum likelihood (ML)

decoder, minimum average error probability, generalized Plotkin bound

-||- _|_ _|_ / __|__ 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: Fri May 27 08:18:35 UTC+8 2016