An elementary derivation of the annihilator polynomial for extremal (2s + 1)-designs

✍ Scribed by M.S. Shrikhande; N.M. Singhi

• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 249 KB
• Volume: 80
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(90)90298-v

✦ Synopsis

Let D be a (2s + 1)-design with parameters (v, k, &+r ). It is known that D has at least s + 1 block intersection numbers x,, x2, . . , xs+,. Suppose now D is an extremal (2~ + l)-design with exactly s + 1 intersection numbers. In this case we give a short proof of the following known result of Delsarte: The s + 1 intersection numbers are roots of a polynomial whose coefficients depend only on the design parameters. Delsarte's result, proved more generally, for designs in Q-polynomial association schemes, uses the notion of the annihilator polynomial. Our proof relies on elementary ideas and part of an algorithm used for decoding BCH codes. * Acknowledges support from Central Michigan University Summer Fellowship award #4-21264. ** This paper was written during a visit to the Ohio State University.