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Generalized perfect arrays and menon difference sets

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Abstract

Given an s 1 × ... × s rinteger-valued array A and a (0, 1) vector z = (z 1, ..., z r), form the array A′ from A by recursively adjoining a negative copy of the current array for each dimension i where z i = 1. A is a generalized perfect array type z if all periodic autocorrelation coefficients of A′ are zero, except for shifts (u 1, ..., u r) where u i, - 0 (mod s i) for all i. The array is perfect if z = (0, ..., 0) and binary if the array elements are all ±1. A nontrivial perfect binary array (PBA) is equivalent to a Menon difference set in an abelian group.

Using only elementary techniques, we prove various construction theorems for generalized perfect arrays and establish conditions on their existence. We show that a generalized PBA whose type is not (0, ..., 0) is equivalent to a relative difference set in an abelian factor group. We recursively construct several infinite families of generalized PBAs, and deduce nonexistence results for generalized PBAs whose type is not (0, ..., 0) from well-known nonexistence results for PBAs. A central result is that a PBA with 22y32u elements and no dimension divisible by 9 exists if and only if no dimension is divisible by 2y+2. The results presented here include and enlarge the set of sizes of all previously known generalized PBAs.

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  1. Hewlett-Packard Laboratories, Filton Road, BS12 6QZ, Stoke Gifford, Bristol, UK

    Jonathan Jedwab

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Communicated by D. Jungnickel

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Jedwab, J. Generalized perfect arrays and menon difference sets. Des Codes Crypt 2, 19–68 (1992). https://doi.org/10.1007/BF00124209

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