MinT - Best Known (81−41, 81, s)-Nets in Base 5

Best Known (81−41, 81, s)-Nets in Base 5

(81−41, 81, 78)-Net over F₅ — Constructive and digital

Digital (40, 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]

(81−41, 81, 85)-Net over F₅ — Digital

Digital (40, 81, 85)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−41, 81, 1283)-Net in Base 5 — Upper bound on s

There is no (40, 81, 1284)-net in base 5, because

1 times m-reduction [i] would yield (40, 80, 1284)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 83 854205 999370 462069 253340 458184 701795 241018 765073 720065 > 5⁸⁰ [i]