The generalized Gilbert-Varshamov bound is implied by Turan's theorem [code construction]
Ludo Tolhuizen · IEEE Transactions on Information Theory · 1997
The generalization of the Gilbert-Varshamov bound due to Gu and Fuja (1993) is a direct consequence of Turan's theorem on the existence of a clique in a graph with many edges. Turan's theorem allows a slight improvement of Gu and Fuja's result. This improved generalized Gilbert-Varshamov bound is in fact equivalent to Turan's theorem.