MinT - Best Known (13−12, 13, s)-Nets in Base 25

Best Known (13−12, 13, s)-Nets in Base 25

(13−12, 13, 27)-Net over F₂₅ — Constructive and digital

Digital (1, 13, 27)-net over F₂₅, using

net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
- Niederreiter sequence [i]

(13−12, 13, 36)-Net over F₂₅ — Digital

Digital (1, 13, 36)-net over F₂₅, using

net from sequence [i] based on digital (1, 35)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 1 and N(F) ≥ 36, using
  - sharp bound on the number of rational points on curves with genus 1 [i]

(13−12, 13, 123)-Net in Base 25 — Upper bound on s

There is no (1, 13, 124)-net in base 25, because

4 times m-reduction [i] would yield (1, 9, 124)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁹, 124, S₂₅, 8), but
  - the linear programming bound shows that MÂ â‰¥ 119725 208199 920654 296875 / 31015 449154 > 25⁹ [i]