In this paper, we re-introduce from our previous work a new family of
nonlinear codes, called weak flip codes, and show that
its subfamily fair weak flip codes belongs to the class of
equidistant codes, satisfying that any two distinct codewords have
identical Hamming distance. It is then noted that the fair weak flip
codes are related to the binary nonlinear Hadamard codes as
both code families maximize the minimum Hamming distance and meet
the Plotkin upper bound under certain blocklengths. Although the fair
weak flip codes have the largest minimum Hamming distance and achieve
the Plotkin bound, we find that these codes are by no means optimal in
the sense of average error probability over binary symmetric channels
(BSC). In parallel, this result implies that the equidistant Hadamard
codes are also nonoptimal over BSCs. Such finding is in contrast to
the conventional code design that aims at the maximization of the
minimum Hamming distance.

The results in this paper are proved by examining the exact error
probabilities of these codes on BSCs, using the column-wise
analysis on the codebook matrix.

-||-   _|_ _|_     /    __|__   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: Mon Apr 22 16:01:21 UTC+8 2013