MinT - Best Known (82−65, 82, s)-Nets in Base 27

Best Known (82−65, 82, s)-Nets in Base 27

(82−65, 82, 96)-Net over F₂₇ — Constructive and digital

Digital (17, 82, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 82, 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]

(82−65, 82, 144)-Net over F₂₇ — Digital

Digital (17, 82, 144)-net over F₂₇, using

t-expansion [i] based on digital (16, 82, 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]

(82−65, 82, 2048)-Net in Base 27 — Upper bound on s

There is no (17, 82, 2049)-net in base 27, because

1 times m-reduction [i] would yield (17, 81, 2049)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 87 323547 038846 779300 900216 351509 802396 875835 697313 003840 412648 167022 104109 127655 976401 932278 895694 654253 689804 258177 > 27⁸¹ [i]