
HP Labs Technical Reports
Generalized Perfect Arrays and Menon Difference Sets
Jedwab, Jonathan
HPL9129
Keyword(s):
Abstract: Using only elementary techniques, we prove various construction theorems for generalized perfect arrays and establish conditions on there existence. We show that a perfect binary array (PBS) is equivalent to a Menon difference set in an abelian group, and 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 PBAs whose type is not (0,...,0) from wellknown 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 devisable by 2y+2. The results presented here include and enlarge the set of sizes of all previously known generalized PBAs.
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