Extending permutation arrays: improving MOLS bounds

Abstract

A permutation array (PA) A is a set of permutations on $Z_n=\{0,1,\dots ,n-1\}$, for some n. A PA A has pairwise Hamming distance at least d, if for every pair of permutations $\sigma$ and $\tau$ in A, there are at least d integers i in $Z_n$ such that $\sigma (i)\ne \tau (i)$. Let M(n, d) denote the maximum number of permutations in any PA with pairwise Hamming distance at least d. Recently considerable effort has been devoted to improving known lower bounds for M(n, d) for all $n>d>3$. We give a partition and extension operation that enables the production of a new PA $A'$ for $M(n+1,d)$ from an existing PA A for $M(n,d-1)$. In particular, this operation allows for improvements for PA’s for $M(q+1,q)$ for powers of prime numbers q, as well as for many other choices of n and d, where n is not a power of a prime. Finally, for prime numbers p, the partition and extension technique provides an asymptotically better lower bound for $M(p+1,p)$ than that given by current knowledge about mutually orthogonal Latin squares. We prove a new asymptotic lower bound for the set of primes p, namely, $M(p+1,p)\ge p^{1.5}/2-O(p)$.