Infinite Families of 3-Designs from Z 4 -Goethals Codes with Block Size 8
Kalle Ranto · SIAM Journal on Discrete Mathematics · 2002
We construct several new families of simple 3-designs from codewords of the $\mathbf Z_4$-Goethals codes. These designs have parameters 3-$(2^m,8,\lambda)$ with odd $m\ge 5$. The smallest design has $\lambda=14(2^m-8)/3$, and the others are corollaries of this and some previously known designs. In the existence proofs we analyze the low-weight codewords and count the number of solutions to certain systems of equations over finite fields.