MinT - Best Known (81−69, 81, s)-Nets in Base 27

Best Known (81−69, 81, s)-Nets in Base 27

(81−69, 81, 96)-Net over F₂₇ — Constructive and digital

Digital (12, 81, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 81, 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]

(81−69, 81, 109)-Net over F₂₇ — Digital

Digital (12, 81, 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]

(81−69, 81, 1196)-Net in Base 27 — Upper bound on s

There is no (12, 81, 1197)-net in base 27, because

1 times m-reduction [i] would yield (12, 80, 1197)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 246995 434645 175137 229770 692049 521386 406376 144007 958323 718647 925868 653477 513452 182318 719725 439101 107016 510922 334897 > 27⁸⁰ [i]