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

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### abstract

• We consider the communication of information in the presence of synchronization errors. Specifically, we consider permutation channels in which a transmitted codeword ${x}=(x_{1},\ldots ,x_{n})$ is corrupted by a permutation $\pi \in S_{n}$ to yield the received word ${y}=(y_{1},\ldots ,y_{n})$ , where $y_{i}=x_{\pi (i)}$ . We initiate the study of worst case (or zero-error) communication over permutation channels that distort the information by applying permutations $\pi$ , which are limited to displacing any symbol by at most $r$ locations, i.e., permutations $\pi$ with weight at most $r$ in the $\ell _\infty$ -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 $r=1$ .

### publication date

• December 1, 2017