MinT - Best Known (10, 10+41, s)-Nets in Base 81

Best Known (10, 10+41, s)-Nets in Base 81

(10, 10+41, 172)-Net over F₈₁ — Constructive and digital

Digital (10, 51, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 51, 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+41, 244)-Net over F₈₁ — Digital

Digital (10, 51, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 51, 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+41, 6119)-Net in Base 81 — Upper bound on s

There is no (10, 51, 6120)-net in base 81, because

1 times m-reduction [i] would yield (10, 50, 6120)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 265782 248166 120696 381646 434606 447261 124060 508989 616349 313631 730813 721361 578530 250536 274117 952001 > 81⁵⁰ [i]