Perfect binary arrays and difference sets

Abstract

A perfect binary array is an r-dimensional array with elements k 1 such that all out-of-phase periodic autocorrelation coefficients are zero. Such an array is equivalent to a Menon difference set in an abelian group. We give recursive constructions for four infinite families of two-dimensional perfect binary arrays, using only elementary methods. Brief outlines of the proofs were previously given by three of the authors. Although perfect binary arrays of the same sizes as two of the families were constructed earlier by Davis, the sizes of the other two families are new.

Generalized perfect arrays and menon difference sets

“…The construction methods outlined in [35] and [59] were proved in detail by Jedwab et al [36], who also showed: Since there exists a GPBA(2, 2) type (0, 0) and simultaneously type (1, 0), Result 4.7 may be recursively applied, starting with the four infinite families of PBA given by (4.1) and (4.2). In this way L/ike et al [43] constructed PBAs in dimensions larger than 2 that are not obtainable from Result 4.2 and (4.1), (4.2) alone.…”

“…We may represent these two-dimensional arrays as In previous papers [2], [30], [35], [36], [37], [58], [59], B i was called perfect (i = 0), rowwise quasiperfect (i = 1), columnwise quasiperfect (i = 2), or doubly quasiperfect (i = 3) provided gBi (Ul, U2) 7 6 0 only if u 1 ~-~ 0 (mod sl) and u 2 ~ 0 (mod s2).…”

“…The result that (i) and (ii) of Theorem 7.3 implies (iii) when r' = 1 and (z, z') = (0, .... 0) was given for the case ml = m2 = 2, Zo = 0 (i.e., Result 4.7) by L/ike et al [43], for the case ml = mE = 3, z0 = 0 by Antweiler et al [2], and for the case A = [++], ml = m2 = 2, z0 = 1 by Jedwab [30]. The cases (m, r) = (2, 0) and (2, 1) of Theorem 7.4 are due respectively to Geramita and Seberry [22] and to Jedwab et al [36]. We next show under what conditions a GPA may be transformed into another of the same type but different size by altering the interleaving of its component arrays.…”

“…Conversely if (i) and (ii) hold then both sides of (7.12) equal 0 and so (iii) holds. The case m = r = 2, z = (0, 0) of Lemma 7.2 was given by Jedwab et al [36]. The next two theorems form the heart of the recursive constructions we shall apply in Section 9.…”

Necessary conditions for menon difference sets

“…One can read [3], [4], [7], and [8] for more on perfect binary arrays. When N is of the form 2~3 b, many constructions exist (see [5], [6], [7], [8] and [14]). Recently, Arasu, et al gave a new construction of Menon difference sets in a class of abelian groups and obtained the following result [1].…”