Weak Flip Codes and their Optimality on the Binary Erasure Channel

This paper investigates fundamental properties of nonlinear binary codes by looking at the codebook matrix not row-wise (codewords), but column-wise. The family of weak flip codes is presented and shown to contain many beautiful properties. In particular the subfamily fair weak flip codes, which goes back to 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 to an arbitrary number of codewords. Based on the column-wise approach, the $r$-wise Hamming distance is introduced as a generalization to the widely used (pairwise) Hamming distance. It is shown that the minimum $r$-wise Hamming distance satisfies a generalized $r$-wise Plotkin bound. The $r$-wise Hamming distance structure of the nonlinear fair weak flip codes is analyzed and shown to be superior to many codes. In particular, it is proven that the fair weak flip codes achieve the $r$-wise Plotkin bound with equality for all $r$. In the second part of the paper, these insights are applied to a binary erasure channel (BEC) with an arbitrary erasure probability. An exact formula for the average error probability of an arbitrary code using maximum likelihood decoding is derived and shown to be expressible using only the $r$-wise Hamming distance structure of the code. For a number of codewords $M\leq4$ and an arbitrary blocklength $n$, the globally optimal codes (in the sense of minimizing the average error probability) are found. For $M=5,6$ and an arbitrary blocklength $n$, the optimal codes are conjectured. For larger $M$, observations regarding the optimal design are presented, e.g., that good codes have a large $r$-wise Hamming distance structure for all $r$. Numerical results validate our code design criteria and show the superiority of our best found nonlinear weak flip codes compared to the best linear codes.