D-optimal weighing designs for n≡−1 mod4 objects and a large number of weighings

✍ Scribed by Bernardo M. Ábrego; Silvia Fernández-Merchant; Michael G. Neubauer; William Watkins

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 327 KB
• Volume: 374
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(03)00614-1

✦ Synopsis

Let M m,n (0, 1) denote the set of all m × n (0, 1)-matrices and let

In this paper we exhibit some new formulas for G(m, n) where n ≡ -1 (mod 4). Specifically, for m = nt + r where 0 r < n, we show that for all sufficiently large t, G(nt + r, n) is a polynomial in t of degree n that depends on the characteristic polynomial of the adjacency matrix of a certain regular graph. Thus the problem of finding G(nt + r, n) for large t is equivalent to finding a regular graph, whose degree of regularity and number of vertices depend only on n and r, with a certain "trace-minimal" property. In particular we determine the appropriate trace-minimal graph and hence the formulas for G(nt + r, n) for n = 11, 15, all r, and all sufficiently large t.