MinT - Best Known (6, 6+69, s)-Nets in Base 81

Best Known (6, 6+69, s)-Nets in Base 81

(6, 6+69, 160)-Net over F₈₁ — Constructive and digital

Digital (6, 75, 160)-net over F₈₁, using

t-expansion [i] based on digital (5, 75, 160)-net over F₈₁, using
- net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
    - a function field by SÃ©mirat [i]

(6, 6+69, 190)-Net over F₈₁ — Digital

Digital (6, 75, 190)-net over F₈₁, using

net from sequence [i] based on digital (6, 189)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 6 and N(F) ≥ 190, using
  - a curve from the manYPoints database [i]

(6, 6+69, 2361)-Net in Base 81 — Upper bound on s

There is no (6, 75, 2362)-net in base 81, because

11 times m-reduction [i] would yield (6, 64, 2362)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 139 459899 586666 572872 011284 532748 715307 115615 541739 159484 989145 632335 736004 348617 776556 338314 777709 997049 945763 062008 027041 > 81⁶⁴ [i]