# Upper Bounds for Chebyshev Permutation Arrays

## TL;DR

This paper improves upper bounds for permutation arrays using the Chebyshev metric by analyzing separable string arrays.

## Contribution

A novel upper bound for P(n,d) is derived using separable arrays of ternary strings and combinatorial methods.

## Key findings

- R(n;k,k) is an upper bound for P(n,n−k) when k ≤ n/2.
- New recursive and combinatorial methods yield improved upper bounds for P(n,d).

## Abstract

We improve on known upper bounds for the size of permutation arrays under the Chebyshev metric, defined as follows. The Chebyshev distance between permutations π and σ on the symbols {1,2,…,n}, denoted by d(π,σ), is max{|πi−σi||1≤i≤n}. For an array A (set) of such permutations, the Chebyshev distance of A, denoted by d(A), is min{d(π,σ)|π,σ∈A,π≠σ}. An array A of such permutations with d(A)=d will be called an (n,d)-PA. Let P(n,d) denote the maximum size of any (n,d)-PA. The function P(n,d) has been the subject of previous research. In this paper, we consider strings on the symbols {0,1,2}, with the 0’s representing low symbols and the 2’s high symbols for the function P(n,d). An array A of such strings of length n is separable if for any two strings in A, there is a position 1≤i≤n such that the ith symbol in one string is 0 and the ith symbol in the other is a 2. The maximum size of a separable array of strings of length n, with a occurrences of the symbol 0 and b occurrences of the symbol 2, is denoted by R(n;a,b). We show that R(n;k,k) is an upper bound for P(n,n−k) when k≤n2. We derive upper bounds for R(n;a,b) by various recursive and combinatorial methods, from which follow upper bounds for the Chebyshev function P(n,d), which improve upon previous such upper bounds in the literature.