True minimum distance conjecture for the even-like LCD BCH code

Let $q$ be odd and $m \geq 2$. For an integer $t$ with $1 \leq t \leq \bar m$, let $\delta=(q^t-1)/2$ and let ${\mathcal{C}}_{(q,n,2\delta,\frac{n}{2}-\delta+1)}$ be the code from Theorem 1. True minimum distance conjecture. The code ${\mathcal{C}}_{(q,n,2\delta,\frac{n}{2}-\delta+1)}$ has true minimum distance $q^t-1$. Numerical experiments support this for $(q,m) \in \{(3,2),(3,3),(3,4),(3,5),(5,2),(7,2)\}$ and $1 \leq t \leq \bar m$; the theorem gives the lower bound $d \geq q^t-1$, while equality is conjectural in general.