An infinite Family of Non-embeddable Hadamard Designs

Abstract

The parameters $2$ - $(2\lambda+2,\lambda+1,\lambda)$ are those of a residual Hadamard $2$ - $(4\lambda+3,2\lambda+1,\lambda)$ design. All $2$ - $(2\lambda+2,\lambda+1,\lambda)$ designs with $\lambda \le 4$ are embeddable. The existence of non-embeddable Hadamard $2$-designs has been determined for the cases $\lambda = 5$, $\lambda = 6$, and $\lambda = 7$. In this paper the existence of an infinite family of non-embeddable $2$ - $(2\lambda+2,\lambda+1,\lambda)$ designs, $\lambda = 3(2^m) - 1, m \ge 1$ is established.