MinT - Best Known (57−48, 57, s)-Nets in Base 81

Best Known (57−48, 57, s)-Nets in Base 81

(57−48, 57, 172)-Net over F₈₁ — Constructive and digital

Digital (9, 57, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 57, 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]

(57−48, 57, 244)-Net over F₈₁ — Digital

Digital (9, 57, 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]

(57−48, 57, 4165)-Net in Base 81 — Upper bound on s

There is no (9, 57, 4166)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 6 083886 754444 381977 586347 050969 216672 436197 382972 083746 986274 793472 405142 042051 706509 961765 789634 442854 437121 > 81⁵⁷ [i]