MinT - Best Known (78−65, 78, s)-Nets in Base 27

Best Known (78−65, 78, s)-Nets in Base 27

(78−65, 78, 96)-Net over F₂₇ — Constructive and digital

Digital (13, 78, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 78, 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]

(78−65, 78, 136)-Net over F₂₇ — Digital

Digital (13, 78, 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]

(78−65, 78, 1351)-Net in Base 27 — Upper bound on s

There is no (13, 78, 1352)-net in base 27, because

1 times m-reduction [i] would yield (13, 77, 1352)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 166 222695 233333 180049 681020 893246 906167 740931 689352 024763 192660 610447 020645 931058 143110 046204 078487 623193 006081 > 27⁷⁷ [i]