Weak Flip Codes and their Optimality on the Binary Erasure Channel

This paper investigates fundamental properties of <italic>nonlinear</italic> binary codes by looking at the codebook matrix not row-wise (codewords), but <italic>column-wise</italic>. The family of <italic>weak flip codes</italic> is presented and shown to contain many beautiful properties. In particular the subfamily <italic>fair weak flip codes</italic>, which goes back to Shannon <italic>et al.</italic> and which was shown to achieve the error exponent with a fixed number of codewords <inline-formula> <tex-math notation="LaTeX">$\mathsf {M}$ </tex-math></inline-formula>, can be seen as a generalization of linear codes to an arbitrary number of codewords. The fair weak flip codes are related to binary nonlinear Hadamard codes. Based on the column-wise approach to the codebook matrix, the <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-<italic>wise Hamming distance</italic> is introduced as a generalization to the well-known and widely used (pairwise) Hamming distance. It is shown that the minimum <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-wise Hamming distance satisfies a <italic>generalized</italic> <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-<italic>wise Plotkin bound</italic>. The <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-wise Plotkin bound with equality for all <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>. In the second part of this paper, these insights are applied to a <italic>binary erasure channel</italic> with an arbitrary erasure probability 0 < <inline-formula> <tex-math notation="LaTeX">$\delta$ </tex-math></inline-formula> < 1. An exact formula for the average error probability of an arbitrary (linear or nonlinear) code using maximum likelihood decoding is derived and shown to be expressible using only the <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-wise Hamming distance structure of the code. For a number of codewords <inline-formula> <tex-math notation="LaTeX">$\mathsf {M}$ </tex-math></inline-formula> satisfying <inline-formula> <tex-math notation="LaTeX">$\mathsf {M}\le 4$ </tex-math></inline-formula> and an arbitrary finite blocklength <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, the globally optimal codes (in the sense of minimizing the average error probability) are found. For <inline-formula> <tex-math notation="LaTeX">$\mathsf {M}$ </tex-math></inline-formula> = 5 or <inline-formula> <tex-math notation="LaTeX">$\mathsf {M}$ </tex-math></inline-formula> = 6 and an arbitrary finite blocklength <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, the optimal codes are conjectured. For larger <inline-formula> <tex-math notation="LaTeX">$\mathsf {M}$ </tex-math></inline-formula>, observations regarding the optimal design are presented, e.g., that good codes have a large <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>-wise Hamming distance structure for all <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>. Numerical results validate our code design criteria and show the superiority of our best found nonlinear weak flip codes compared with the best linear codes.