MinT - Best Known (57−45, 57, s)-Nets in Base 27

Best Known (57−45, 57, s)-Nets in Base 27

(57−45, 57, 96)-Net over F₂₇ — Constructive and digital

Digital (12, 57, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 57, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(57−45, 57, 109)-Net over F₂₇ — Digital

Digital (12, 57, 109)-net over F₂₇, using

net from sequence [i] based on digital (12, 108)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 12 and N(F) ≥ 109, using
  - a curve from the manYPoints database [i]

(57−45, 57, 1520)-Net in Base 27 — Upper bound on s

There is no (12, 57, 1521)-net in base 27, because

1 times m-reduction [i] would yield (12, 56, 1521)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 143 343901 471894 708645 875454 600655 916678 299553 913244 945200 345526 594077 937638 370409 > 27⁵⁶ [i]