MinT - Best Known (87−57, 87, s)-Nets in Base 5

Best Known (87−57, 87, s)-Nets in Base 5

(87−57, 87, 51)-Net over F₅ — Constructive and digital

Digital (30, 87, 51)-net over F₅, using

t-expansion [i] based on digital (22, 87, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(87−57, 87, 58)-Net over F₅ — Digital

Digital (30, 87, 58)-net over F₅, using

net from sequence [i] based on digital (30, 57)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 30 and N(F) ≥ 58, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(87−57, 87, 375)-Net in Base 5 — Upper bound on s

There is no (30, 87, 376)-net in base 5, because

1 times m-reduction [i] would yield (30, 86, 376)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 297054 832482 129795 087097 235498 058293 499819 703410 296077 978625 > 5⁸⁶ [i]