MinT - Best Known (9, 9+57, s)-Nets in Base 81

Best Known (9, 9+57, s)-Nets in Base 81

(9, 9+57, 172)-Net over F₈₁ — Constructive and digital

Digital (9, 66, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 66, 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+57, 244)-Net over F₈₁ — Digital

Digital (9, 66, 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+57, 3790)-Net in Base 81 — Upper bound on s

There is no (9, 66, 3791)-net in base 81, because

1 times m-reduction [i] would yield (9, 65, 3791)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 11266 746491 644276 182525 435349 672412 519478 275061 690909 448028 839918 423609 671974 554318 074405 114434 832345 209672 035194 035038 815041 > 81⁶⁵ [i]