Improved Upper Bounds for Strength-3 Covering Arrays over Six Symbols via Mixed Short and Regular PSL(2, 5) Orbits

For an integer $t \ge 1$, the covering array number $\mathrm{CAN}(t, k, v)$ is the minimum number of rows $N$ in a strength-$t$ covering array $\mathrm{CA}(N; t, k, v)$ with $k$ columns over $v$ symbols. Identifying $\{0, 1, 2, 3, 4, 5\}$ with $\mathbb{P}^1(\mathbb{F}_5) = \mathbb{F}_5 \,{\cup}\, \{\infty\}$ under the diagonal action of $G = \mathrm{PSL}(2, 5)$ ($|G| = 60$), we record that every $G$-orbit on $\mathbb{P}^1(\mathbb{F}_5)^k$ ($k \ge 1$) has size in $\{6, 30, 60\}$ (all three occurring for $k \ge 3$), where each $30$-row orbit is generated canonically by a non-constant binary starter $b \in \{0, \infty\}^k$ of stabilizer order $2$ in $G$, placing a complement-closed binary pair across the $15$ symbol pairs and projecting bijectively onto $30$-element two-symbol $G$-orbits on every non-constant $3$-column triple. Jointly selecting $1$ constant starter, $5$ binary $\{0, \infty\}^{12}$ starters, and $5$ regular $60$-row starters yields an explicit group-orbit covering array $\mathrm{CA}(456; 3, 12, 6)$ specified by $11$ vectors, improving $\mathrm{CAN}(3, 12, 6) \le 463$ by $7$ rows. Applying conflict-driven tabu row deletion to mixed-$(30, 60)$ $\mathrm{PSL}(2, 5)$ orbit seeds further establishes $\mathrm{CAN}(3, 10, 6) \le 366$ ($< 380$, $-14$ rows) and $\mathrm{CAN}(3, 11, 6) \le 418$ ($< 420$, $-2$ rows), with every row in all three arrays verified to be essential. Substituting $\mathrm{CAN}(3, 10, 6) \le 366$ into Chateauneuf--Kreher doubling with $\mathrm{CAN}(2, 10, 6) \le 48$ then yields $\mathrm{CAN}(3, 19, 6) \le \mathrm{CAN}(3, 20, 6) \le 606$ ($< 620$, $-14$ rows).