MinT - Best Known (68−51, 68, s)-Nets in Base 27

Best Known (68−51, 68, s)-Nets in Base 27

(68−51, 68, 96)-Net over F₂₇ — Constructive and digital

Digital (17, 68, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 68, 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]

(68−51, 68, 144)-Net over F₂₇ — Digital

Digital (17, 68, 144)-net over F₂₇, using

t-expansion [i] based on digital (16, 68, 144)-net over F₂₇, using
- net from sequence [i] based on digital (16, 143)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 16 and N(F) ≥ 144, using
    - a curve from the manYPoints database [i]

(68−51, 68, 2670)-Net in Base 27 — Upper bound on s

There is no (17, 68, 2671)-net in base 27, because

1 times m-reduction [i] would yield (17, 67, 2671)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 799904 296267 719806 390016 948517 917724 357002 129902 686328 645391 531749 834140 740801 081880 173333 703383 > 27⁶⁷ [i]