Coding for the ℓ_∞-limited permutation channel

In this work we consider the communication of information in the presence of synchronization errors. Specifically, we consider permutation channels in which a transmitted codeword x = (x₁,⋯, x_n) is corrupted by a permutation π ϵ S_n to yield the received word y = (y₁,⋯, y_n) where y_i = x_π(i). We initiate the study of worst case (or zero error) communication over permutation channels that distort the information by applying permutations π which are limited to displacing any symbol by at most r locations, i.e. permutations π with weight at most r in the ℓ_∞-metric. We present direct and recursive constructions, as well as bounds on the rate of such channels for binary and general alphabets. Specific attention is given to the case of r = 1.