Abstract

An extension from the pairwise Hamming distance to the \(r\)-wise Hamming distance is presented. It can be used to fully characterize the maximum-likelihood decoding (MLD) error of an arbitrary code over the binary erasure channel (BEC). By noting that good codes always have large minimum \(r\)-wise Hamming distances for all \(r\), a new design criterion for a code is introduced: the minimum \(r\)-wise Hamming distance. We then prove an upper bound for the minimum \(r\)-wise Hamming distance of an arbitrary code, called the generalized Plotkin bound, and provide a class of (nonlinear) codes that achieve the bound for every \(r\).