Nonlinear codes outperform the best linear codes on the binary erasure channel

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 < δ < 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
.