Heng et al.'s conjecture on infinite families of 1-MDS codes

Throughout this section, let $q=2^h$ with $h\geq 3$, let $\alpha$ be a primitive element of $\mathbb{F}_q$, and write $\alpha_i=\alpha^i$ for $1\leq i\leq q-1$. For $3\leq k\leq q-2$, define

$$M_k=\left[\begin{array}{ccccc} 1&1&\cdots&1&1\\ \alpha_1&\alpha_2&\cdots&\alpha_{q-2}&\alpha_{q-1}\\ \alpha_1^2&\alpha_2^2&\cdots&\alpha_{q-2}^2&\alpha_{q-1}^2\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \alpha_1^{k-2}&\alpha_2^{k-2}&\cdots&\alpha_{q-2}^{k-2}&\alpha_{q-1}^{k-2}\\ \alpha_1^k&\alpha_2^k&\cdots&\alpha_{q-2}^k&\alpha_{q-1}^k \end{array}\right],$$

and let $\mathcal{C}_k$ be the $q$-ary linear code generated by the rows of $M_k$. A linear code with parameters $[n,k,d]_q$ is called 1-MDS when its minimum distance is one less than the Singleton bound, namely $d=n-k$.

Heng et al.'s conjecture. For each $3\leq k\leq q-2$, the linear code $\mathcal{C}_k$ is a $1$-MDS $[q-1,k,q-k-1]_q$ code, and the minimum-weight codewords of both $\mathcal{C}_k$ and its dual $\mathcal{C}_k^{\perp}$ support $2$-designs.

This conjecture proposes an infinite family of near-MDS-type codes arising from a matrix whose full-rank minors are governed by zero-sum subsets of $\mathbb{F}_q^*$. It remains open in the supplied source, which attributes the conjecture to Heng et al.