Extended (2, 4)-designs

In this paper, the concept of an extended (2, 4)-design is introduced. An extended (2, 4)-design is a pair (X, $\text{B}$) where X is a finite set and $\text{B}$ is a collection of 4-tuples of not necessarily distinct elements of X, such that every pair of not necessarily distinct elements of X is contained in exactly one member of $\text{B}$. It is shown that an extended (2,4)-design of order n exists for every positive integer n except n = 6, 8 and 9. Several inequivalent designs of order n are obtained.