Constructing Constant Composition Codes via Distance-Increasing Mappings

Abstract

A distance-preserving mapping is a one-to-one function $f$ from $p$-ary vectors of length $m$ to $q$-ary vectors of length $n$ such that any two distinct $p$-ary vectors are mapped to two different $q$-ary vectors with an equal or greater Hamming distance. A distance-increasing mapping is a special distance-preserving mapping which strictly increases the distance by at least one if the distance of two distinct input vectors is less than the length of the output vectors. A constant composition code over a $k$-ary alphabet has the property that the numbers of occurrences of the $k$ symbols within a codeword are fixed for each codeword. One of the most important applications of distance-preserving mappings and distance-increasing mappings is to construct constant composition codes, of which the permutation codes are a special subclass. There are two results in this paper. First, we propose a swap-based distance-increasing mapping from binary vectors to quaternary constant composition vectors. Second, we prove that it is impossible to construct any swap-based distance-preserving mappings from binary vectors to ternary constant composition vectors under the swap model that we defined.