Maximizing Hamming Distance in Contraction of Permutation Arrays
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A permutation array A is set of permutations on a finite set Ω, say of size n. Given distinct permutations π, σ∈Ω, we let hd(π, σ) = |{ x∈Ω: π(x) σ(x) }|, called the Hamming distance between σ and τ. Now let hd(A) = min{ hd(π, σ): π, σ∈ A }. For positive integers n and d with d< n we let M(n,d) be the maximum number of permutations in any array A satisfying hd(A) ≥ d. There is an extensive literature on the function M(n,d), motivated in part by applications to error correcting codes for message transmission over power lines. In this paper we consider the case where q is a prime power with q≡ 1 (mod 3). For this case, we give lower bounds for M(q-1,q-3) if q≥ 7, and when q is odd for M(q,q-3) if q≥ 13. These bounds are based on a contraction operation applied to the permutation groups AGL(1,q) and PGL(2,q). We obtain additional lower bounds on M(n,d) for a finite number of pairs (n,d) by applying the contraction operation to the Mathieu groups.
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