Good permutation codes based on the shuffle-exchange network

Abstract

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 ∈ ℕ, we present a collection of N = N(n) = (n!)^Ω(1) pairwise far apart permutations {π_i: [n] → [n]}_i∈[N] and a polynomial-time algorithm that on input i ∈ [N] outputs an explicit description of π_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 constant composition codes on smaller alphabets, where in these codes 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.