Generalized perfect arrays and Menon difference sets
Given an \(s_ 1\times\cdots\times s_ r\) integer-valued array \(A\) and a \((0,1)\) vector \({\mathbf z}=(z_ 1,\dots,z_ r)\), 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 \({\mathbf z}\) if all periodic autocorrelation coefficients of \(A'\) are zero, except for shifts \((u_ 1,\dots,u_ r)\) where \(u_ i\equiv 0\pmod{s_ i}\) for all \(i\). The array is perfect if \({\mathbf z}=(0,\dots,0)\) and binary if the array elements are all \(\pm 1\). A non- trivial 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 \((0,\dots,0)\) 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 \((0,\dots,0)\) from well-known nonexistence results for PBAs. A central result is that a PBA with \(2^{2y} 3^{2u}\) elements and no dimension divisible by 9 exists if and only if no dimension is divisible by \(2^{y+2}\). The results presented here include and enlarge the set of sizes of all previously known generalized PBAs.
scientific article; zbMATH DE number 95047
scientific article