MinT - Best Known (99−39, 99, s)-Nets in Base 25

Best Known (99−39, 99, s)-Nets in Base 25

(99−39, 99, 334)-Net over F₂₅ — Constructive and digital

Digital (60, 99, 334)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (9, 22, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by GarcÃa and Quoos [i]
2. digital (9, 28, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅ (see above)
3. digital (10, 49, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]

(99−39, 99, 2764)-Net over F₂₅ — Digital

Digital (60, 99, 2764)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−39, 99, 5363285)-Net in Base 25 — Upper bound on s

There is no (60, 99, 5363286)-net in base 25, because

1 times m-reduction [i] would yield (60, 98, 5363286)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 99568 556439 426652 862430 515693 723819 870801 048882 568590 194244 247765 623023 076240 817374 565630 449759 391670 356975 544594 059698 201680 981779 869425 > 25⁹⁸ [i]