A new family of nonlinear codes, called weak flip codes,
is presented and is shown to contain many beautiful properties. In
particular, the subfamily of fair weak flip codes can be seen
as a generalization of linear codes. Different from linear codes that
only exist for a number of codewords M being an integer-power
of 2, the fair weak flip code can be defined for an arbitrary
M. It is then noted that the fair weak flip codes are related
to binary nonlinear Hadamard codes: both code families maximize
the minimum Hamming distance and meet the Plotkin bound. However,
while the binary nonlinear Hadamard codes have only been shown to
possess good Hamming-distance properties, the fair weak flip codes are
proven to be globally optimal (in the sense of minimizing the error
probability) among all linear or nonlinear codes for the binary
erasure channel (BEC) for many values of the blocklength n and
for M <= 6. For M > 6, similar optimality results are

The results in this paper are founded on a new powerful tool for the
analysis and generation of block codes: the column-wise
approach to 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: Wed Jun 27 18:27:49 UTC+8 2012