MinT - Best Known (166−41, 166, s)-Nets in Base 4

Best Known (166−41, 166, s)-Nets in Base 4

(166−41, 166, 1028)-Net over F₄ — Constructive and digital

Digital (125, 166, 1028)-net over F₄, using

4² times duplication [i] based on digital (123, 164, 1028)-net over F₄, using
- trace code for nets [i] based on digital (0, 41, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(166−41, 166, 1676)-Net over F₄ — Digital

Digital (125, 166, 1676)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−41, 166, 256538)-Net in Base 4 — Upper bound on s

There is no (125, 166, 256539)-net in base 4, because

1 times m-reduction [i] would yield (125, 165, 256539)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2187 337343 700876 038684 898460 611728 795855 578526 921628 605808 018410 624588 339932 316618 932379 054515 741846 > 4¹⁶⁵ [i]