An all-degree bound for odd-order abelian Cayley graphs of diameter two

Let A_odd(d) be the maximum order of an odd-order abelian Cayley graph of degree at most d and diameter at most two. Applying the line-product lift of Zhang and Ge to the ternary radius-two covering-code parameters of Cock and Östergård gives the provisional bound A_odd(d) ≥ (12/25)d² − O(d^(99/59)) for all sufficiently large d. An elementary argument gives a deficit O(d²/ln d). More generally, a radius-two linear covering-code family over a fixed finite field of odd order with lower limiting density μ gives a graph liminf coefficient of at least 1/(2μ).