binary codes by looking at the codebook matrix not row-wise

(codewords), but

codes

properties. In particular the subfamily

which goes back to Berlekamp, Gallager, and Shannon and which was

shown to achieve the error exponent with a fixed number of codewords

arbitrary number of codewords. The fair weak flip codes are related

to binary nonlinear Hadamard codes.

Based on the column-wise approach to the codebook matrix, the

to the well-known and widely used (pairwise) Hamming distance. It is

shown that the minimum

distance structure of the nonlinear fair weak flip codes is analyzed

and shown to be superior to many codes. In particular, it is proven

that the fair weak flip codes achieve the

with equality for all

In the second part of the paper, these insights are applied to a

probability

probability of an arbitrary (linear or nonlinear) code using maximum

likelihood decoding is derived and shown to be expressible using

only the

number of codewords

arbitrary finite blocklength

sense of minimizing the average error probability) are found. For

observations regarding the optimal design are presented, e.g., that

good codes have a large

the superiority of our best found nonlinear weak flip codes compared

to the best linear codes.

bound, maximum likelihood (ML) decoder, minimum average error

probability, optimal nonlinear code design,

weak flip codes.

-||- _|_ _|_ / __|__ 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 Feb 4 13:50:44 CET 2020