Asymptotic counting exponent for partial Hadamard matrices

Upper bound, last-listed row

| $3$ | [Davis2026] | Current best upper bound: the cubic-regime result proves $N_{n,4t} \sim A_{n,4t}$ for $t/n^3 \to \infty$ and shows that a nonvanishing correction survives when $t = \Theta n^3$ with large fixed $\Theta$, so the asymptotics change at the cubic scale. [Davis2026-change-cubic] [Davis2026-open-below3] |

exact pinned row

| $3$ | <a href="#Davis2026">[Davis2026]</a> | Current best upper bound: the cubic-regime result proves $N_{n,4t} \sim A_{n,4t}$ for $t/n^3 \to \infty$ and shows that a nonvanishing correction survives when $t = \Theta n^3$ with large fixed $\Theta$, so the asymptotics change at the cubic scale. <a href="#Davis2026-change-cubic">[Davis2026-change-cubic]</a> <a href="#Davis2026-open-below3">[Davis2026-open-below3]</a> |

tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $1$ | [DL2010] | Admissible exponents cannot be below $1$: for any $\alpha<1$, the regime $t/n^\alpha \to \infty$ still includes widths with $4t<n$, where an $n \times 4t$ partial Hadamard matrix cannot exist. [DL2010-linear-obstruction] |

exact pinned row

| $1$ | <a href="#DL2010">[DL2010]</a> | Admissible exponents cannot be below $1$: for any $\alpha<1$, the regime $t/n^\alpha \to \infty$ still includes widths with $4t<n$, where an $n \times 4t$ partial Hadamard matrix cannot exist. <a href="#DL2010-linear-obstruction">[DL2010-linear-obstruction]</a> |

tracked since 2026-07-24 — no movement seen yet