MinT - Best Known (30, 30+45, s)-Nets in Base 5

Best Known (30, 30+45, s)-Nets in Base 5

(30, 30+45, 51)-Net over F₅ — Constructive and digital

Digital (30, 75, 51)-net over F₅, using

t-expansion [i] based on digital (22, 75, 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]

(30, 30+45, 58)-Net over F₅ — Digital

Digital (30, 75, 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]

(30, 30+45, 492)-Net in Base 5 — Upper bound on s

There is no (30, 75, 493)-net in base 5, because

1 times m-reduction [i] would yield (30, 74, 493)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 5482 633552 956655 470866 256969 446457 028381 971732 504905 > 5⁷⁴ [i]