Coding for the $\boldsymbol \ell _\infty $ -Limited Permutation Channel

We consider the communication of information in the presence of synchronization errors. Specifically, we consider permutation channels in which a transmitted codeword x = (x
₁
, ... , x
_n
) is corrupted by a permutation π ∈ S
_n
to yield the received wordy = (y
₁
, . . . , 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.