Nonvanishing determinant conjecture for the Atrahasis code construction

Let $n$, $k$, $d$, and $t$ be integers such that $n-1\geq d\geq k\geq 2$ and $d\leq t(d-k+1)$. Let

$$\alpha=\binom{(t-1)(d-k+1)}{t-1}.$$

The main theorem establishes that, when $\alpha\leq 3003$, there exists an $(n,k,d,\alpha)$-MSR code over some sufficiently large field. Atrahasis-code conjecture. The main theorem holds for all $\alpha$; equivalently, for every such choice of parameters, there exists an $(n,k,d,\alpha)$-MSR code over some sufficiently large field. The claimed extension is motivated by computations showing no counterexample for $\alpha\leq 3003$ and by the authors' belief that the determinant underlying both proofs is nonzero for all $\alpha$.