MinT - Best Known (9, 9+52, s)-Nets in Base 81

Best Known (9, 9+52, s)-Nets in Base 81

(9, 9+52, 172)-Net over F₈₁ — Constructive and digital

Digital (9, 61, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 61, 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+52, 244)-Net over F₈₁ — Digital

Digital (9, 61, 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+52, 3948)-Net in Base 81 — Upper bound on s

There is no (9, 61, 3949)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 263 011622 941236 027108 726869 480452 720082 158985 607084 741807 976004 718221 861762 906134 072254 422732 528598 447735 408420 213921 > 81⁶¹ [i]