New Bounds on the Size of Permutation Codes With Minimum Kendall $\tau$-distance of Three

TL;DR

This paper establishes tighter upper bounds on the maximum size of permutation codes with minimum Kendall tau-distance of three, using group theory and integer programming techniques, improving previous bounds for various cases.

Contribution

The authors derive improved upper bounds on permutation code sizes with Kendall tau-distance 3, notably reducing bounds for prime and specific small values of n.

Findings

Upper bounds for prime p ≥ 11 improved to (p-1)! - ⌈p/3⌉ + 2

Specific bounds for n=6,7,11,13,14,15,17 reduced by 3,3,9,11,1,1,4 respectively

Method combines group theory and integer programming to tighten bounds

Abstract

We study $P(n,3)$, the size of the largest subset of the set of all permutations $S_n$ with minimum Kendall $\tau$-distance $3$. Using a combination of group theory and integer programming, we reduced the upper bound of $P(p,3)$ from $(p-1)!-1$ to $(p-1)!-\lceil\frac{p}{3}\rceil+2\leq (p-1)!-2$ for all primes $p\geq 11$. In special cases where $n$ is equal to $6,7,11,13,14,15$ and $17$ we reduced the upper bound of $P(n,3)$ by $3,3,9,11,1,1$ and $4$, respectively.