MinT - Best Known (76−67, 76, s)-Nets in Base 81

Best Known (76−67, 76, s)-Nets in Base 81

(76−67, 76, 172)-Net over F₈₁ — Constructive and digital

Digital (9, 76, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 76, 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]

(76−67, 76, 244)-Net over F₈₁ — Digital

Digital (9, 76, 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]

(76−67, 76, 3562)-Net in Base 81 — Upper bound on s

There is no (9, 76, 3563)-net in base 81, because

1 times m-reduction [i] would yield (9, 75, 3563)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 137725 222852 227649 072965 498896 057194 805347 817985 441637 800114 543556 470929 904781 012577 039209 821590 157368 112872 220296 462247 075617 101833 108712 433521 > 81⁷⁵ [i]