MinT - Best Known (70−33, 70, s)-Nets in Base 5

Best Known (70−33, 70, s)-Nets in Base 5

(70−33, 70, 72)-Net over F₅ — Constructive and digital

Digital (37, 70, 72)-net over F₅, using

t-expansion [i] based on digital (31, 70, 72)-net over F₅, using
- net from sequence [i] based on digital (31, 71)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
    - an explicitly constructive algebraic function field [i]

(70−33, 70, 97)-Net over F₅ — Digital

Digital (37, 70, 97)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(70−33, 70, 1745)-Net in Base 5 — Upper bound on s

There is no (37, 70, 1746)-net in base 5, because

1 times m-reduction [i] would yield (37, 69, 1746)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 700425 754229 405373 324028 413595 727538 605112 365825 > 5⁶⁹ [i]