Large Cayley digraphs of given degree and diameter

We construct a family of Cayley digraphs of degree d , diameter k and order k ⌊ d / 2 ⌋ k for any d ≥ 4 and k ≥ 3 . We also present a collection of bipartite Cayley digraphs of order at least ( k − 1 ) ⌊ d / 2 ⌋ k − 1 for any degree d ≥ 4 and diameter k ≥ 4 . For sufficiently large d and k , our digraphs are the largest known Cayley digraphs of degree d and diameter k , where k ≠ d − 1 or d , and our bipartite digraphs are the largest known bipartite Cayley digraphs.