MinT - Best Known (39−29, 39, s)-Nets in Base 81

Best Known (39−29, 39, s)-Nets in Base 81

(39−29, 39, 172)-Net over F₈₁ — Constructive and digital

Digital (10, 39, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 39, 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]

(39−29, 39, 244)-Net over F₈₁ — Digital

Digital (10, 39, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 39, 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]

(39−29, 39, 11436)-Net in Base 81 — Upper bound on s

There is no (10, 39, 11437)-net in base 81, because

1 times m-reduction [i] would yield (10, 38, 11437)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 3 332485 359862 376159 927973 438944 060747 067104 434673 474486 469541 000013 788641 > 81³⁸ [i]