MinT - Best Known (130−89, 130, s)-Nets in Base 5

Best Known (130−89, 130, s)-Nets in Base 5

(130−89, 130, 78)-Net over F₅ — Constructive and digital

Digital (41, 130, 78)-net over F₅, using

t-expansion [i] based on digital (38, 130, 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]

(130−89, 130, 80)-Net over F₅ — Digital

Digital (41, 130, 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]

(130−89, 130, 451)-Net in Base 5 — Upper bound on s

There is no (41, 130, 452)-net in base 5, because

1 times m-reduction [i] would yield (41, 129, 452)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 546153 448545 862913 824000 685292 009843 611826 044422 404546 698106 198434 340709 755711 620408 775425 > 5¹²⁹ [i]