MinT - Best Known (3, 53, s)-Nets in Base 25

Best Known (3, 53, s)-Nets in Base 25

(3, 53, 52)-Net over F₂₅ — Constructive and digital

Digital (3, 53, 52)-net over F₂₅, using

net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
  - a function field by GarcÃa and Quoos [i]

(3, 53, 56)-Net over F₂₅ — Digital

Digital (3, 53, 56)-net over F₂₅, using

net from sequence [i] based on digital (3, 55)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 56, using
  - fields by Serre with genus 3 or 4 [i]

(3, 53, 274)-Net in Base 25 — Upper bound on s

There is no (3, 53, 275)-net in base 25, because

4 times m-reduction [i] would yield (3, 49, 275)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁴⁹, 275, S₂₅, 46), but
  - the linear programming bound shows that MÂ â‰¥ 6 430313 750241 332147 612039 871646 663038 243180 749329 590988 917770 832955 511650 652624 666690 826416 015625 / 20311 023951 484409 636688 674357 > 25⁴⁹ [i]