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This paper investigates fundamental properties of nonlinear binary codes bylooking at the codebook matrix not row-wise (codewords), but column-wise. Thefamily of weak flip codes is presented and shown to contain many beautifulproperties. In particular the subfamily fair weak flip codes, which goes backto Berlekamp, Gallager, and Shannon and which was shown to achieve the errorexponent with a fixed number of codewords $M$, can be seen as a generalizationof linear codes to an arbitrary number of codewords. Based on the column-wiseapproach, the $r$-wise Hamming distance is introduced as a generalization tothe 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 isanalyzed and shown to be superior to many codes. In particular, it is proventhat the fair weak flip codes achieve the $r$-wise Plotkin bound with equalityfor all $r$. In the second part of the paper, these insights are applied to abinary erasure channel (BEC) with an arbitrary erasure probability. An exactformula for the average error probability of an arbitrary code using maximumlikelihood 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 thesense of minimizing the average error probability) are found. For $M=5,6$ andan arbitrary blocklength $n$, the optimal codes are conjectured. For larger$M$, observations regarding the optimal design are presented, e.g., that goodcodes have a large $r$-wise Hamming distance structure for all $r$. Numericalresults validate our code design criteria and show the superiority of our bestfound nonlinear weak flip codes compared to the best linear codes.

Weak Flip Codes and their Optimality on the Binary Erasure Channel