MinT - Best Known (69−55, 69, s)-Nets in Base 27

Best Known (69−55, 69, s)-Nets in Base 27

(69−55, 69, 96)-Net over F₂₇ — Constructive and digital

Digital (14, 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]

(69−55, 69, 136)-Net over F₂₇ — Digital

Digital (14, 69, 136)-net over F₂₇, using

t-expansion [i] based on digital (13, 69, 136)-net over F₂₇, using
- net from sequence [i] based on digital (13, 135)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 13 and N(F) ≥ 136, using
    - a curve from the manYPoints database [i]

(69−55, 69, 1677)-Net in Base 27 — Upper bound on s

There is no (14, 69, 1678)-net in base 27, because

1 times m-reduction [i] would yield (14, 68, 1678)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 21 587009 219038 036681 908045 680765 600263 352809 058162 609304 715586 389098 502569 912435 161136 066072 997537 > 27⁶⁸ [i]