MinT - Best Known (41, 41+51, s)-Nets in Base 5

Best Known (41, 41+51, s)-Nets in Base 5

(41, 41+51, 78)-Net over F₅ — Constructive and digital

Digital (41, 92, 78)-net over F₅, using

t-expansion [i] based on digital (38, 92, 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]

(41, 41+51, 80)-Net over F₅ — Digital

Digital (41, 92, 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]

(41, 41+51, 872)-Net in Base 5 — Upper bound on s

There is no (41, 92, 873)-net in base 5, because

1 times m-reduction [i] would yield (41, 91, 873)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4066 508929 220449 989946 126001 782222 364612 960509 773399 236741 567941 > 5⁹¹ [i]