MinT - Best Known (81−79, 81, s)-Nets in Base 25

Best Known (81−79, 81, s)-Nets in Base 25

(81−79, 81, 28)-Net over F₂₅ — Constructive and digital

Digital (2, 81, 28)-net over F₂₅, using

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

(81−79, 81, 46)-Net over F₂₅ — Digital

Digital (2, 81, 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]

(81−79, 81, 72)-Net in Base 25 — Upper bound on s

There is no (2, 81, 73)-net in base 25, because

12 times m-reduction [i] would yield (2, 69, 73)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁶⁹, 73, S₂₅, 67), but
  - the linear programming bound shows that MÂ â‰¥ 24178 564222 846123 668546 399721 917528 895562 151649 914153 732205 158997 548011 257094 913162 291049 957275 390625 / 7242 > 25⁶⁹ [i]