MinT - Best Known (9, 9+51, s)-Nets in Base 81

Best Known (9, 9+51, s)-Nets in Base 81

(9, 9+51, 172)-Net over F₈₁ — Constructive and digital

Digital (9, 60, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 60, 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+51, 244)-Net over F₈₁ — Digital

Digital (9, 60, 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+51, 4048)-Net in Base 81 — Upper bound on s

There is no (9, 60, 4049)-net in base 81, because

1 times m-reduction [i] would yield (9, 59, 4049)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 40109 253784 794061 132438 264972 495048 913008 210592 338340 822300 301211 497743 051691 942234 349606 124734 927284 052280 098001 > 81⁵⁹ [i]