MinT - Best Known (18, 18+21, s)-Nets in Base 9

Best Known (18, 18+21, s)-Nets in Base 9

(18, 18+21, 74)-Net over F₉ — Constructive and digital

Digital (18, 39, 74)-net over F₉, using

t-expansion [i] based on digital (17, 39, 74)-net over F₉, using
- net from sequence [i] based on digital (17, 73)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
    - a function field by GarcÃa and Quoos [i]

(18, 18+21, 75)-Net over F₉ — Digital

Digital (18, 39, 75)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(9³⁹, 75, F₉, 2, 21) (dual of [(75, 2), 111, 22]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(9³⁷, 74, F₉, 2, 21) (dual of [(74, 2), 111, 22]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(2;F,126P) [i] based on function field F/F₉ with g(F) = 16 and N(F) ≥ 74, using
    - a curve from the manYPoints database [i]

(18, 18+21, 2387)-Net in Base 9 — Upper bound on s

There is no (18, 39, 2388)-net in base 9, because

1 times m-reduction [i] would yield (18, 38, 2388)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 827750 174840 228049 733530 615939 271873 > 9³⁸ [i]