MinT - Best Known (69−57, 69, s)-Nets in Base 27

Best Known (69−57, 69, s)-Nets in Base 27

(69−57, 69, 96)-Net over F₂₇ — Constructive and digital

Digital (12, 69, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 69, 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]

(69−57, 69, 109)-Net over F₂₇ — Digital

Digital (12, 69, 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]

(69−57, 69, 1286)-Net in Base 27 — Upper bound on s

There is no (12, 69, 1287)-net in base 27, because

1 times m-reduction [i] would yield (12, 68, 1287)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 21 873921 151930 105217 967075 387607 342834 978088 704040 586986 603709 247559 159979 995731 645450 274576 824105 > 27⁶⁸ [i]