MinT - Best Known (81−27, 81, s)-Nets in Base 4

Best Known (81−27, 81, s)-Nets in Base 4

(81−27, 81, 195)-Net over F₄ — Constructive and digital

Digital (54, 81, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 27, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(81−27, 81, 270)-Net over F₄ — Digital

Digital (54, 81, 270)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−27, 81, 9567)-Net in Base 4 — Upper bound on s

There is no (54, 81, 9568)-net in base 4, because

1 times m-reduction [i] would yield (54, 80, 9568)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 462388 421838 861483 205360 022952 628875 910050 315045 > 4⁸⁰ [i]