MinT - Best Known (81−39, 81, s)-Nets in Base 5

Best Known (81−39, 81, s)-Nets in Base 5

(81−39, 81, 78)-Net over F₅ — Constructive and digital

Digital (42, 81, 78)-net over F₅, using

t-expansion [i] based on digital (38, 81, 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]

(81−39, 81, 100)-Net over F₅ — Digital

Digital (42, 81, 100)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−39, 81, 1724)-Net in Base 5 — Upper bound on s

There is no (42, 81, 1725)-net in base 5, because

1 times m-reduction [i] would yield (42, 80, 1725)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 82 863799 150566 558088 402417 480638 506532 373158 205461 785581 > 5⁸⁰ [i]