MinT - Best Known (68−47, 68, s)-Nets in Base 8

Best Known (68−47, 68, s)-Nets in Base 8

(68−47, 68, 65)-Net over F₈ — Constructive and digital

Digital (21, 68, 65)-net over F₈, using

t-expansion [i] based on digital (14, 68, 65)-net over F₈, using
- net from sequence [i] based on digital (14, 64)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
    - the Suzuki function field over F₈ [i]

(68−47, 68, 76)-Net over F₈ — Digital

Digital (21, 68, 76)-net over F₈, using

t-expansion [i] based on digital (20, 68, 76)-net over F₈, using
- net from sequence [i] based on digital (20, 75)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 20 and N(F) ≥ 76, using
    - a curve from the manYPoints database [i]

(68−47, 68, 561)-Net in Base 8 — Upper bound on s

There is no (21, 68, 562)-net in base 8, because

1 times m-reduction [i] would yield (21, 67, 562)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 292649 693055 188335 023808 145304 655057 462526 196221 967033 840624 > 8⁶⁷ [i]