MinT - Best Known (10, 10+57, s)-Nets in Base 81

Best Known (10, 10+57, s)-Nets in Base 81

(10, 10+57, 172)-Net over F₈₁ — Constructive and digital

Digital (10, 67, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 67, 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]

(10, 10+57, 244)-Net over F₈₁ — Digital

Digital (10, 67, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 67, 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]

(10, 10+57, 4437)-Net in Base 81 — Upper bound on s

There is no (10, 67, 4438)-net in base 81, because

1 times m-reduction [i] would yield (10, 66, 4438)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 915146 253673 587964 554440 013225 066562 424268 466247 867072 111245 858709 988829 485315 867866 400602 575727 396020 896631 423365 004091 208321 > 81⁶⁶ [i]