HP Labs Technical Reports

Generalized Perfect Arrays and Menon Difference Sets

Jedwab, Jonathan



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 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 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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