MinT - Best Known (98−57, 98, s)-Nets in Base 5

Best Known (98−57, 98, s)-Nets in Base 5

(98−57, 98, 78)-Net over F₅ — Constructive and digital

Digital (41, 98, 78)-net over F₅, using

t-expansion [i] based on digital (38, 98, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(98−57, 98, 80)-Net over F₅ — Digital

Digital (41, 98, 80)-net over F₅, using

net from sequence [i] based on digital (41, 79)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 41 and N(F) ≥ 80, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(98−57, 98, 725)-Net in Base 5 — Upper bound on s

There is no (41, 98, 726)-net in base 5, because

1 times m-reduction [i] would yield (41, 97, 726)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 65 332097 605612 029754 475728 041365 184817 009404 748422 367717 662263 212225 > 5⁹⁷ [i]