MinT - Best Known (78−63, 78, s)-Nets in Base 27

Best Known (78−63, 78, s)-Nets in Base 27

(78−63, 78, 96)-Net over F₂₇ — Constructive and digital

Digital (15, 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]

(78−63, 78, 136)-Net over F₂₇ — Digital

Digital (15, 78, 136)-net over F₂₇, using

t-expansion [i] based on 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]

(78−63, 78, 1699)-Net in Base 27 — Upper bound on s

There is no (15, 78, 1700)-net in base 27, because

1 times m-reduction [i] would yield (15, 77, 1700)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 166 122809 272430 931060 244164 748025 239563 462628 427105 355734 790643 126626 049632 797534 966058 029639 244088 733837 438609 > 27⁷⁷ [i]