Abstract
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.