MinT - Best Known (9, 9+37, s)-Nets in Base 81

Best Known (9, 9+37, s)-Nets in Base 81

(9, 9+37, 172)-Net over F₈₁ — Constructive and digital

Digital (9, 46, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 46, 172)-net over F₈₁, using
- net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
    - a function field by SÃ©mirat [i]

(9, 9+37, 244)-Net over F₈₁ — Digital

Digital (9, 46, 244)-net over F₈₁, using

net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
  - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

(9, 9+37, 5566)-Net in Base 81 — Upper bound on s

There is no (9, 46, 5567)-net in base 81, because

1 times m-reduction [i] would yield (9, 45, 5567)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 76 330674 408698 066938 649694 794188 261789 425491 758980 015370 016811 803007 342468 781698 282081 > 81⁴⁵ [i]