MinT - Best Known (69−39, 69, s)-Nets in Base 5

Best Known (69−39, 69, s)-Nets in Base 5

(69−39, 69, 51)-Net over F₅ — Constructive and digital

Digital (30, 69, 51)-net over F₅, using

t-expansion [i] based on digital (22, 69, 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]

(69−39, 69, 58)-Net over F₅ — Digital

Digital (30, 69, 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]

(69−39, 69, 615)-Net in Base 5 — Upper bound on s

There is no (30, 69, 616)-net in base 5, because

1 times m-reduction [i] would yield (30, 68, 616)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 344215 936000 700943 245918 363289 035384 445150 181025 > 5⁶⁸ [i]