MinT - Best Known (69−41, 69, s)-Nets in Base 81

Best Known (69−41, 69, s)-Nets in Base 81

(69−41, 69, 370)-Net over F₈₁ — Constructive and digital

Digital (28, 69, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 69, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(69−41, 69, 500)-Net over F₈₁ — Digital

Digital (28, 69, 500)-net over F₈₁, using

t-expansion [i] based on digital (26, 69, 500)-net over F₈₁, using
- net from sequence [i] based on digital (26, 499)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 26 and N(F) ≥ 500, using
    - a curve from the manYPoints database [i]

(69−41, 69, 319927)-Net in Base 81 — Upper bound on s

There is no (28, 69, 319928)-net in base 81, because

1 times m-reduction [i] would yield (28, 68, 319928)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 5983 907153 855548 176239 724267 764808 634219 409701 728839 234089 889364 654014 944075 880739 394360 764424 378436 378657 376435 636152 583715 980801 > 81⁶⁸ [i]