3-designs from Z4-Goethals-like codes and variants of cyclotomic polynomials

Abstract

We construct a family of simple 3-(2m, 8, 14(2m - 8)/3) designs, with odd m ≥ 5, from all ℤ4-Goethals-like codes G??k with k = 21 and l ≥ 1. In addition, these designs imply also the existence of the other design families constructed from the ℤ4-Goethals codes G1 by Ranto. In the existence proofs we count the number of solutions to certain systems of equations over finite fields arid use Dickson polynomials and variants of cyclotomic polynomials and identities connecting them. © Springer-Verlag Berlin Heidelberg 2006.