MinT - Best Known (68−45, 68, s)-Nets in Base 25

Best Known (68−45, 68, s)-Nets in Base 25

(68−45, 68, 148)-Net over F₂₅ — Constructive and digital

Digital (23, 68, 148)-net over F₂₅, using

t-expansion [i] based on digital (19, 68, 148)-net over F₂₅, using
- net from sequence [i] based on digital (19, 147)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 19 and N(F) ≥ 148, using
    - a function field by GarcÃa and Quoos [i]

(68−45, 68, 176)-Net over F₂₅ — Digital

Digital (23, 68, 176)-net over F₂₅, using

net from sequence [i] based on digital (23, 175)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 23 and N(F) ≥ 176, using
  - an algebraic function field by Teo [i]

(68−45, 68, 6812)-Net in Base 25 — Upper bound on s

There is no (23, 68, 6813)-net in base 25, because

1 times m-reduction [i] would yield (23, 67, 6813)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 4605 919152 241461 198274 042410 597138 769041 542732 948521 003496 449202 692469 382824 924336 234988 816145 > 25⁶⁷ [i]