MinT - Best Known (68−53, 68, s)-Nets in Base 27

Best Known (68−53, 68, s)-Nets in Base 27

(68−53, 68, 96)-Net over F₂₇ — Constructive and digital

Digital (15, 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]

(68−53, 68, 136)-Net over F₂₇ — Digital

Digital (15, 68, 136)-net over F₂₇, using

t-expansion [i] based on 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]

(68−53, 68, 1967)-Net in Base 27 — Upper bound on s

There is no (15, 68, 1968)-net in base 27, because

1 times m-reduction [i] would yield (15, 67, 1968)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 804296 091780 707870 612215 799232 898330 130064 101478 809059 604336 013924 095106 043205 937601 765988 259105 > 27⁶⁷ [i]