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 P GL(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 ≡ 2 (mod 3) such that n + 1 is a prime power.

Constructing permutation arrays using partition and extension

“…It contains 38 · 37 · 36 = 50, 616 permutations with Hamming distance at least 36, giving M (38, 36) ≥ 50, 616. Using the coset method [2], we found five cosets of P GL(2, 37) in S 38 , with Hamming distance 34 from P GL(2, 37) (see Table 8). The cosets are defined by the coset representatives α, β, γ, δ and θ: Note that for all i, j, (1 ≤ i < j ≤ 6), hd(M i ) = 36 and hd(M i , M j ) ≥ 34.…”

“…The operation of contraction on a PA Y on Z n+1 with Hamming distance d + 1 results in new PA Y on Z n . As with the coset method, if Y is a good PA for M (n + 1, d), Y could exhibit a new lower bound for either M (n, d − 2) or M (n, d − 3), depending on conditions described in [2].…”

“…Our coset search techniques are discussed in Section 6.3. n d r NEW Origin of Blocks (see Table 8 (54, 48)) 70 63 2 1, 503, 462 P P GL(2, 67) and 2 cosets (see M (68, 61)) Table 6: M (n, d) lower bounds obtained using parallel partition and extension (Theorem 6 and 7). The blocks used by these theorems were obtained by the coset method [2] (see Table 8). Columns: r denotes the number of new symbols, NEW denotes the new new bound.…”

New lower bounds for permutation arrays using contraction

“…As we want to consider permutations on F q (not P 1 (F q )) we need to eliminate occurrences of the symbol ∞ using an operation called contraction [1]. If W (∞) = ∞, then we can simply eliminate the symbol ∞, which of course makes no new agreements.…”

“…Permutation arrays (PAs) with large Hamming distance have been the subject of many recent papers with applications in the design of error correcting codes. New lower bounds for the size of such permutation arrays are given, for example, in [1,2,3,4,5,6,7,12,15,14,19,20,22].…”