Conjectured exact minimum distances for D-sequence cyclic codes

Let $m=2^l e>2$, where $2\nmid e$ and $l\geq1$, and let $\mathcal{C}_{\mathcal{D},i}$ denote the cyclic code associated with the sequence family $\mathcal{D}$. Let $h$ be the parameter used in the paper's definition of these codes, and let $d(\mathcal{C})$ denote minimum distance. The D-sequence code minimum-distance conjecture. If $l\geq2$ and $e>1$, then

$$d(\mathcal{C}_{\mathcal{D},1})=2^{\frac{m}{2}}+4$$

for even $h$, while

$$d(\mathcal{C}_{\mathcal{D},1})=2^{\frac{m}{2}}+2$$

for odd $h$. If $l=1$, then

$$d(\mathcal{C}_{\mathcal{D},0})=2^{\frac{m}{2}}+2$$

for $h=1$, and

$$d(\mathcal{C}_{\mathcal{D},0})=2^{\frac{m}{2}}$$

for $h=2$. The asserted values are further numerical predictions for the exact minimum distances of these cyclic codes; the supplied excerpt gives no resolution beyond the conjectural status.