MinT - Best Known (9, 9+68, s)-Nets in Base 81

Best Known (9, 9+68, s)-Nets in Base 81

(9, 9+68, 172)-Net over F₈₁ — Constructive and digital

Digital (9, 77, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 77, 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+68, 244)-Net over F₈₁ — Digital

Digital (9, 77, 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+68, 3535)-Net in Base 81 — Upper bound on s

There is no (9, 77, 3536)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 901 064201 951060 191073 333390 448098 200568 152255 134531 510615 176702 519282 313978 745127 565269 766676 330012 991691 873447 382109 527228 423942 476431 645902 072321 > 81⁷⁷ [i]