Constructing permutation arrays from groups

Abstract

Let M(n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M(n, d). We compute the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1, q) and PGL(2, q) with Frobenius maps to obtain new, improved lower bounds for M(n, d). We give new randomized algorithms. We give better lower bounds for M(n, d) also using new theorems concerning the contraction operation. For example, we prove a quadratic lower bound for $M(n,n-2)$ for all $n\equiv 2 \pmod 3$ such that $n+1$ is a prime power.

Appendix

Finite Fields Let $n=p^k$ be a prime power. There is a field $\mathbb {F}_n$ with n elements, unique up to isomorphism. We consider groups over the field $\mathbb {F}_n$.

Groups The group AGL(1, n) consists of the affine linear transformations

$$\begin{aligned} AGL(1,n) = \{ax + b | a,b\in \mathbb {F}_n, a\not =0\}, \end{aligned}$$

where the group operation is function composition. This group is sharply $2-$transitive and has $n(n-1)$ elements.

Denote the symbols of $Z_{n+1}$ by $0,1,2,\dots ,n-1,\infty$. The permutations of PGL(2, n) are $g:x\rightarrow \frac{ax+b}{cx+d}$ on $Z_{n+1}$ such that $a,b,c,d\in GF(n), ad\ne bc, g(\infty )=a/c,g(-d/c)=\infty$ if $c\ne 0$ and $g(\infty )=\infty$ if $c=0$. Then $|PGL(2,n)|=(n+1)n(n-1)$.

Recall that $n=p^k$. The group of affine semilinear polynomials $A\Gamma L(1,n)$ arises as a semidirect product of AGL(1, n) with a cyclic group of order k. It is generated by iteratively composing the Frobenius automorphism $x^p$ with the elements of AGL(1, n). Equivalently,

$$\begin{aligned} A\Gamma L(1,n) = \{ax^{p^i}+b \mid a,b\in \mathbb {F}_n, a\not =0, 0\le i < k\} \end{aligned}$$

This group has $kn(n-1)$ elements.

The group of projective semilinear polynomials $P\Gamma L(2,n)$ arises as a semidirect product of PGL(2, n) with a cyclic group of order k generated by the Frobenius automorphism. This group has $k(n+1)n(n-1)$ elements.