Large Cayley digraphs of given degree and diameter

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