Abstract

Based on a new way of designing codes using codebook columns, a family
of nonlinear codes, called weak flip codes, is presented
and shown to contain many beautiful properties. In particular the
subfamily fair weak flip codes, which goes back to definitions
by Berlekamp, Gallager, and Shannon and which was shown to achieve the
error exponent with a fixed number of codewords M, can be
seen as a generalization of linear codes. 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 codes (including both
linear and nonlinear ones) for the binary erasure channel (BEC) for
many values of the blocklength n and for a number of codewords
M satisfying M <= 6. For the performance analysis, an
extension of pairwise Hamming distance, the r-wise Hamming distance,
is proposed and found to be a key to the understanding of the codes'
performance.


Keywords

Binary erasure channel (BEC), channel capacity, finite blocklength,
weak flip codes, fair weak flip codes, maximum likelihood (ML)
decoder, minimum average error probability, optimal codes.



-||-   _|_ _|_     /    __|__   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: Thu Oct 2 08:42:39 UTC+8 2014