MinT - Best Known (9, 9+39, s)-Nets in Base 81

Best Known (9, 9+39, s)-Nets in Base 81

(9, 9+39, 172)-Net over F₈₁ — Constructive and digital

Digital (9, 48, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 48, 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]

(9, 9+39, 244)-Net over F₈₁ — Digital

Digital (9, 48, 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]

(9, 9+39, 5204)-Net in Base 81 — Upper bound on s

There is no (9, 48, 5205)-net in base 81, because

1 times m-reduction [i] would yield (9, 47, 5205)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 501458 835837 830248 953067 779436 572986 916495 775770 826732 971327 925682 059009 010710 020861 447601 > 81⁴⁷ [i]