These are the parameters for Hadamard 2-designs. Let $H$ be an $n\times n$ Hadamard matrix, where $n=4m$, normalized so that all entries in the first row are equal to 1. Each row apart from the first gives a partition of $\{1,\ldots,n\}$ into two sets of size $n/2$, hence we get $2n-2$ blocks of size $n/2$. The resulting incidence structure is a 2-design: because the columns of $H$ are pairwise orthogonal, any two columns differ in exactly $n/2$ positions and so any two points lie in exactly $(n-2)/2$ blocks.

It is easy to see that any such design gives rise to a Hadamard matrix, so the designs exist if and only if a Hadamard matrix of order $n$ exists.

Also these designs are actually 3-designs.

These are the parameters for Hadamard 2-designs. Let $H$ be an $n\times n$ Hadamard matrix, where $n=4m$, normalized so that all entries in the first row are equal to 1. Each row apart from the first gives a partition of $\{1,\ldots,n\}$ into two sets of size $n/2$, hence we get $2n-2$ blocks of size $n/2$. The resulting incidence structure is a 2-design: because the columns of $H$ are pairwise orthogonal, any two columns differ in exactly $n/2$ positions and so any two points lie in exactly $(n-2)/2$ blocks.

Also the designs constructed from Hadamard matrices as above are 3-designs, and it can be show that any 3-design with these parameters arises in this way.