MinT - Best Known (85−41, 85, s)-Nets in Base 5

Best Known (85−41, 85, s)-Nets in Base 5

(85−41, 85, 78)-Net over F₅ — Constructive and digital

Digital (44, 85, 78)-net over F₅, using

t-expansion [i] based on digital (38, 85, 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]

(85−41, 85, 103)-Net over F₅ — Digital

Digital (44, 85, 103)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−41, 85, 1775)-Net in Base 5 — Upper bound on s

There is no (44, 85, 1776)-net in base 5, because

1 times m-reduction [i] would yield (44, 84, 1776)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 51803 201603 642281 308239 939026 068174 776892 127701 687435 823105 > 5⁸⁴ [i]