MinT - Best Known (11, 11+57, s)-Nets in Base 81

Best Known (11, 11+57, s)-Nets in Base 81

(11, 11+57, 172)-Net over F₈₁ — Constructive and digital

Digital (11, 68, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 68, 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+57, 244)-Net over F₈₁ — Digital

Digital (11, 68, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 68, 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+57, 5193)-Net in Base 81 — Upper bound on s

There is no (11, 68, 5194)-net in base 81, because

1 times m-reduction [i] would yield (11, 67, 5194)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 73 926325 055484 300203 015069 594527 011266 720848 834067 938916 205621 195167 568896 076689 023823 922911 126735 110217 066716 293435 781242 162561 > 81⁶⁷ [i]