Hadamard 2-Designs from Frobenius Groups Z_v : Z_k

We present a construction of Hadamard 2-designs using affine Frobenius groups of the form Z_v : Z_k . By applying the Key-Moori method to these groups under specific arithmetical conditions—namely, when v = 4λ + 3 and k = 2λ + 1 are both prime—we prove that the resulting symmetric 1-designs are actually Hadamard 2-designs with parameters 2−(4λ + 3, 2λ + 1, λ). This construction provides an explicit, group-theoretic approach to generating an infinite family of Hadamard designs, demonstrating the continued utility of algebraic methods in combinatorial design theory. We discuss computational aspects of the construction and provide examples for small parameter values.