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

For an integer t ≥ 1, the covering array number CAN(t, k, v) is the minimum number of rows N in a strength-t covering array CA(N; t, k, v) with k columns over v symbols. Identifying \0, 1, 2, 3, 4, 5\ with P¹(F₅) = F₅ \,∪\, \∞\ under the diagonal action of G = PSL(2, 5) ($|G| = 60$), we record that every G-orbit on P¹(F₅)ᵏ (k ≥ 1) has size in \6, 30, 60\ (all three occurring for k ≥ 3), where each $30$-row orbit is generated canonically by a non-constant binary starter b ∈ \0, ∞\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, ∞\¹² starters, and $5$ regular $60$-row starters yields an explicit group-orbit covering array CA(456; 3, 12, 6) specified by $11$ vectors, improving CAN(3, 12, 6) ≤ 463 by $7$ rows. Applying conflict-driven tabu row deletion to mixed-$(30, 60)$ PSL(2, 5) orbit seeds further establishes CAN(3, 10, 6) ≤ 366 ($< 380$, $-14$ rows) and CAN(3, 11, 6) ≤ 418 ($< 420$, $-2$ rows), with every row in all three arrays verified to be essential. Substituting CAN(3, 10, 6) ≤ 366 into Chateauneuf--Kreher doubling with CAN(2, 10, 6) ≤ 48 then yields CAN(3, 19, 6) ≤ CAN(3, 20, 6) ≤ 606 ($< 620$, $-14$ rows).