Circulant weighing matrices

Abstract

Algebraic techniques are employed to obtain necessary conditions for the existence of certain circulant w eighing matrices. As an application we rule out the existence of many circulant w eighing matrices.We study orders n = s 2 + s + 1, for 10 s 25. These orders correspond to the numb e r o f p o i n ts in a projective plane of order s.

Nonexistence of CW(154,36) and CWArasu1, Nabavi2 2011Discrete Mathematics4000View full textAdd to dashboardCite

“…To prove that A (2) 1 is a {0, ±1}-element, consider g −1 A and proceed similarly. Now we prove that A 1 and A 0 are both nonzero.…”

“…Hence, A ′ is a CW(2n, k 2 ). (Observe that ψ only (possibly) changes the sign of the coefficients of A; (2) and…”

“…But (a 7 + 5) 2 ≤ 12 forces a 7 + 5 = 3, implying that a 7 = −2. Hence, at this point, we have (a 1 + 5a 6 + 5a 8 ) + 9 = ±6 and (a 1 + 5a 6 + 5a 8 ) 2 = 9. We can see that the only possibility is a 1 + 5a 6 + 5a 8 = −3.…”

“…By [2], there is no CW(85, 64); hence, in order to make use of Theorem 2.10, we just need to find a complete set of nonequivalent ICW 2 (85, 64) matrices. Let A be an ICW(85, 64).…”

Nonexistence of CW(110, 100)

“…Arasu and Gutman [2] have filled in many of the missing entries of Strassler [8]. In this paper, we prove the non-existence of CW (110, 100) using algebraic methods.…”