MinT - Best Known (57−39, 57, s)-Nets in Base 81

Best Known (57−39, 57, s)-Nets in Base 81

(57−39, 57, 370)-Net over F₈₁ — Constructive and digital

Digital (18, 57, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 57, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(57−39, 57, 41786)-Net in Base 81 — Upper bound on s

There is no (18, 57, 41787)-net in base 81, because

1 times m-reduction [i] would yield (18, 56, 41787)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 75027 699201 990494 291542 646199 039192 877790 579030 223345 921277 615136 331730 490526 856801 276284 976681 224992 271441 > 81⁵⁶ [i]