MinT - Best Known (11, 11+48, s)-Nets in Base 81

Best Known (11, 11+48, s)-Nets in Base 81

(11, 11+48, 172)-Net over F₈₁ — Constructive and digital

Digital (11, 59, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 59, 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]

(11, 11+48, 244)-Net over F₈₁ — Digital

Digital (11, 59, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 59, 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]

(11, 11+48, 6013)-Net in Base 81 — Upper bound on s

There is no (11, 59, 6014)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 39975 992284 445297 883159 665728 242575 267667 582896 064240 070295 203334 016719 880116 489892 885476 441315 096042 693209 498881 > 81⁵⁹ [i]