New lower bounds for permutation arrays using contraction

Sergey Bereg, Zevi Miller, Luis Gerardo Mojica, Linda Morales, Ivan Hal Sudborough

Abstract: A permutation array A is a set of permutations on a finite set $$\Omega $$ Ω , say of size n. Given distinct permutations $$\pi , \sigma \in \Omega $$ π , σ ∈ Ω , we let $$hd(\pi , \sigma ) = |\{ x\in \Omega : \pi (x) \ne \sigma (x) \}|$$ h d ( π , σ ) = | { x ∈ Ω : π ( x ) ≠ σ ( x ) } | , called the Hamming distance between $$\pi $$ π and $$\sigma $$ σ . Now let $$hd(A) =$$ h d ( A ) = min $$\{ hd(\pi , \sigma ): \pi , \sigma \in A \}$$ { h d ( π , σ ) : π , σ ∈ A } . For positive integers n and d with $$d\le n$$ d ≤ n we let M(n, d) be the maximum number of permutations in any array A satisfying $$hd(A) \ge d$$ h d ( A ) ≥ d . There is an extensive literature on the function M(n, d), motivated in part by suggested applications to error correcting codes for message transmission over power lines. A basic fact is that if a permutation group G is sharply k-transitive on a set of size $$n\ge k$$ n ≥ k , then $$M(n,n-k+1) = |G|$$ M ( n , n - k + 1 ) = | G | . Motivated by this we consider the permutation groups AGL(1, q) and PGL(2, q) acting sharply 2-transitively on GF(q) and sharply 3-transitively on $$GF(q)\cup \{\infty \}$$ G F ( q ) ∪ { ∞ } respectively. Applying a contraction operation to these groups, we obtain the following new lower bounds for prime powers q satisfying $$q\equiv 1$$ q ≡ 1 (mod 3). 1. $$M(q-1,q-3)\ge (q^{2} - 1)/2$$ M ( q - 1 , q - 3 ) ≥ ( q 2 - 1 ) / 2 for q odd, $$q\ge 7$$ q ≥ 7 , 2. $$M(q-1,q-3)\ge (q-1)(q+2)/3$$ M ( q - 1 , q - 3 ) ≥ ( q - 1 ) ( q + 2 ) / 3 for q even, $$q\ge 8$$ q ≥ 8 , 3. $$M(q,q-3)\ge Kq^{2}log(q)$$ M ( q , q - 3 ) ≥ K q 2 l o g ( q ) for some constant $$K>0$$ K > 0 if q is odd. These results resolve a case left open in a previous paper (Bereg et al. in Des Codes Cryptogr 86(5):1095–1111, 2018), where it was shown that $$M(q-1, q-3) \ge q^{2} - q$$ M ( q - 1 , q - 3 ) ≥ q 2 - q and $$M(q,q-3) \ge q^{3} - q$$ M ( q , q - 3 ) ≥ q 3 - q for all prime powers q such that $$q\not \equiv 1$$ q ≢ 1 (mod 3). We also obtain lower bounds for M(n, d) for a finite number of exceptional pairs n, d, by applying this contraction operation to the sharply 4 and 5-transitive Mathieu groups.