MinT - Best Known (68−59, 68, s)-Nets in Base 81

Best Known (68−59, 68, s)-Nets in Base 81

(68−59, 68, 172)-Net over F₈₁ — Constructive and digital

Digital (9, 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]

(68−59, 68, 244)-Net over F₈₁ — Digital

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]

(68−59, 68, 3729)-Net in Base 81 — Upper bound on s

There is no (9, 68, 3730)-net in base 81, because

1 times m-reduction [i] would yield (9, 67, 3730)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 74 300798 844390 162460 719166 595452 414660 669077 167986 324722 380576 237050 827551 148998 720711 043655 650838 065714 940874 055564 972802 829601 > 81⁶⁷ [i]