Two-block covering arrays from elliptic curves and curves of genus two

For a prime power q, Reed–Solomon codes give orthogonal arrays of index one with q+1 columns, so CAN(t, k, q) = q^t for t ≤ k ≤ q+1. Two hyperplanes of a Riemann–Roch space on a curve over F_q give covering arrays with more columns. On an elliptic curve E, take the functions of L((t+1)O) whose value at a point of degree two lies in F_q or in F_q·z, for a fixed z ∈ F_{q²} ∖ F_q. They form a CA(2q^t − q^(t−1); t, n, q) on the n rational points of E, for every q ≥ 5 and every 2 ≤ t ≤ n. This extends a construction of our earlier preprint from n ≥ t+2 to all t ≤ n, and n can be taken equal to the largest number N_q(1) of points for every q ≥ 5. For genus g the blocks are L(D − Q1) and L(D − Q2) for two divisors of degree g. They cover all t-sets for every t and every D when a condition on the Jacobian holds that involves neither t nor D. For genus two the condition reads [Q1 − Q2] ≠ [R − R′] for all rational points R, R′. A character-sum argument shows that such pairs exist on every curve of genus two once q ≥ 23, and this gives a CA(2q^t − q^(t−2); t, n, q) on its n rational points. A pencil column adds one more column for odd q. The arrays are those of the restricted two-row covering perfect hash families of Colbourn and Lanus, which were found by computer. An orthogonal array over the next prime field limits their advantage to a finite range of q, which we determine for t ≤ 6. For v ≤ 25, 102 of the arrays given by the theorems lie below the best known sizes recorded in the ca-tools table on 29 October 2026, 39 of them on curves of genus two. Files: paper_C.pdf is the paper. supplement_C.zip holds the supplementary data. It contains the certificates of the elliptic and genus-two arrays, the programs that check them and their logs, the explicit arrays CA(33271; 5, 16, 7) and CA(232897; 6, 16, 7), the curves with N_q(2) points for q = 23, 29, 31, the programs for the bounds, the records comparison with copies of the ca-tools table, and the Lean 4 sources (see its README.md). SHA256SUMS.txt lists the checksums.