Two-row covering perfect hash families from Singer cycles and an improved table of covering arrays

A covering perfect hash family CPHF(n;k,q,t) yields a covering array CA(n(q^t-1)+1; t, k, q), and a CA(n(q^t-q)+q; t, k, q) when every row misses a hyperplane. For every prime power q we give an explicit CPHF(2; 2q+2, q, 4). Its first row consists of two orbits of the third symmetric power of a Singer cycle of PG(1,q) (two twisted cubics), its second row of two orbits of a cyclic group of order q+1 on two conics in a pencil. The proof reduces the critical 4-sets to an exact determinant factorisation and a one-line conjugation argument. For every q a suitable member of the family has both rows missing a plane; for odd q this rests on a Kloosterman-sum estimate over Fq^3. Consequently CAN(4, 2q+2, q-j) <= 2q^4 - q - 2j for every prime power q and 0 <= j <= q-2; for j = 0 this improves the current table for q in {13, 16, 17, 19, 23, 25}. We also give a closed-form criterion for the three-parameter family of such arrays, a second infinite family for odd q, and computer-found two-row certificates of a related Singer type for strengths five and six. Finally we list 237 covering arrays that improve the ca-tools table and are not claimed in the companion paper: 87 new constructions (among them two-block arrays from elliptic curves), 98 obtained from arrays by symbol fusion and deletion of rows or columns, 3 compositions with Reed-Solomon perfect hash families, 32 post-optimised published arrays and 17 reconstructions of published constructions missing from the table. The 183 explicit arrays were checked exhaustively by a checker written independently of the construction code. This record is the preprint (PDF). The arrays, certificates, verification programs and Lean 4 sources it refers to are in the data record (doi:10.5281/zenodo.23787055); the companion preprint A is doi:10.5281/zenodo.23787051.