Abstract

The exact value of the average error probability of an arbitrary code (linear or nonlinear) using maximum likelihood
decoding is studied on binary erasure channels (BECs) with arbitrary erasure probability \(0<\delta<1\). The family of the
fair linear codes, which are equivalent to a concatenation of several Hadamard linear codes, is proven to perform
better (in the sense of average error probability with respect to maximum-likelihood decoding) than all other linear codes
for many values of the blocklength n and for a dimension k = 3. It is then noted that the family of fair linear codes and
the family of fair nonlinear weak flip codes both maximize the minimum Hamming distance under certain blocklengths.
However, the fair nonlinear weak flip codes actually outperform the fair linear codes, i.e., linearity and global optimality
cannot be simultaneously achieved for the number of codewords being \(M=2^3\).

Keywords

Binary erasure channel, generalized Plotkin bound, optimal nonlinear channel coding, r-wise Hamming distance,
weak flip codes.