MinT - Best Known (66−55, 66, s)-Nets in Base 81

Best Known (66−55, 66, s)-Nets in Base 81

(66−55, 66, 172)-Net over F₈₁ — Constructive and digital

Digital (11, 66, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 66, 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]

(66−55, 66, 244)-Net over F₈₁ — Digital

Digital (11, 66, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 66, 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]

(66−55, 66, 5354)-Net in Base 81 — Upper bound on s

There is no (11, 66, 5355)-net in base 81, because

1 times m-reduction [i] would yield (11, 65, 5355)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 11274 768202 654438 512712 424343 224157 845780 217488 966965 201998 102126 457039 977792 205951 561471 492152 110939 506689 855316 804377 582801 > 81⁶⁵ [i]