MinT - Best Known (52−41, 52, s)-Nets in Base 81

Best Known (52−41, 52, s)-Nets in Base 81

(52−41, 52, 172)-Net over F₈₁ — Constructive and digital

Digital (11, 52, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 52, 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]

(52−41, 52, 244)-Net over F₈₁ — Digital

Digital (11, 52, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 52, 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]

(52−41, 52, 7626)-Net in Base 81 — Upper bound on s

There is no (11, 52, 7627)-net in base 81, because

1 times m-reduction [i] would yield (11, 51, 7627)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 21 569682 908202 496242 179967 539438 146483 275280 504149 876825 788671 528393 616165 671500 467226 044649 771201 > 81⁵¹ [i]