MinT - Best Known (68−55, 68, s)-Nets in Base 27

Best Known (68−55, 68, s)-Nets in Base 27

(68−55, 68, 96)-Net over F₂₇ — Constructive and digital

Digital (13, 68, 96)-net over F₂₇, using

t-expansion [i] based on digital (11, 68, 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]

(68−55, 68, 136)-Net over F₂₇ — Digital

Digital (13, 68, 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]

(68−55, 68, 1483)-Net in Base 27 — Upper bound on s

There is no (13, 68, 1484)-net in base 27, because

1 times m-reduction [i] would yield (13, 67, 1484)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 805724 010301 419921 176379 804888 606784 630074 096624 752933 433959 730553 506639 584530 804159 446288 080049 > 27⁶⁷ [i]