Constructing Large Families of Pairwise Far Permutations: Good Permutation Codes Based on the Shuffle-Exchange Network

Webpage for a paper by Oded Goldreich and Avi Wigderson


We consider the problem of efficiently constructing an as large as possible family of permutations such that each pair of permutations are far part (i.e., disagree on a constant fraction of their inputs). Specifically, for every $n\in\N$, we present a collection of $N=3DN(n)=3D(n!)^{\Omega(1)}$ pairwise far apart permutations $\{\pi_i:[n]\to[n]\}_{i\in[N]}$ and a polynomial-time algorithm that on input $i\in[N]$ outputs an explicit description of $\pi_i$.

From a coding theoretic perspective, we construct permutation codes of constant relative distance and constant rate along with efficient encoding (and decoding) algorithms. This construction is easily extended to produce codes on smaller alphabets in which every codeword is balanced; namely, each symbol appears the same number of times.

Our construction combines routing on the Shuffle-Exchange network with any good binary error correcting code. Specifically, we uses codewords of a good binary code in order to determine the switching instructions in the Shuffle-Exchange network.

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Additioanl grant acknowledgement: This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 819702).

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