MinT - Best Known (57−37, 57, s)-Nets in Base 81

Best Known (57−37, 57, s)-Nets in Base 81

(57−37, 57, 370)-Net over F₈₁ — Constructive and digital

Digital (20, 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]

(57−37, 57, 81752)-Net in Base 81 — Upper bound on s

There is no (20, 57, 81753)-net in base 81, because

1 times m-reduction [i] would yield (20, 56, 81753)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 75020 429831 679731 388591 822149 857982 968373 734210 230106 058393 445320 391072 602829 044257 477733 039628 521129 519521 > 81⁵⁶ [i]