MinT - Best Known (42−40, 42, s)-Nets in Base 25

Best Known (42−40, 42, s)-Nets in Base 25

(42−40, 42, 28)-Net over F₂₅ — Constructive and digital

Digital (2, 42, 28)-net over F₂₅, using

net from sequence [i] based on digital (2, 27)-sequence over F₂₅, using
- Niederreiter sequence [i]

(42−40, 42, 46)-Net over F₂₅ — Digital

Digital (2, 42, 46)-net over F₂₅, using

net from sequence [i] based on digital (2, 45)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 2 and N(F) ≥ 46, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(42−40, 42, 189)-Net in Base 25 — Upper bound on s

There is no (2, 42, 190)-net in base 25, because

1 times m-reduction [i] would yield (2, 41, 190)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁴¹, 190, S₂₅, 39), but
  - the linear programming bound shows that MÂ â‰¥ 804876 707714 587286 349584 675793 078503 147626 179270 446300 506591 796875 000000 / 388 540779 321991 > 25⁴¹ [i]