Asymptotic coincidence of Hamming bounds for unrestricted data-syndrome codes

Let $d$ be the minimum distance and let $n$ denote the block length. The Hamming bound for non-degenerate data-syndrome codes is the bound in equation (HBnondeg), while unrestricted data-syndrome codes satisfy the bound in equation (HBdeg).

Asymptotic Hamming-bound conjecture. For any $d$ there exists $n(d)$ such that for $n\geq n(d)$ the Hamming bound in equation (HBnondeg) holds for unrestricted data-syndrome codes.

The authors report that the two bounds coincide for $n\geq 36$ when $d=7$, and observe the same behavior for other values of $d$. The conjecture asserts that this eventual coincidence holds for every fixed minimum distance.