MinT - Best Known (219−54, 219, s)-Nets in Base 4

Best Known (219−54, 219, s)-Nets in Base 4

(219−54, 219, 1028)-Net over F₄ — Constructive and digital

Digital (165, 219, 1028)-net over F₄, using

1 times m-reduction [i] based on digital (165, 220, 1028)-net over F₄, using
- trace code for nets [i] based on digital (0, 55, 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]

(219−54, 219, 2137)-Net over F₄ — Digital

Digital (165, 219, 2137)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−54, 219, 278372)-Net in Base 4 — Upper bound on s

There is no (165, 219, 278373)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 709822 393461 008853 890005 526987 528370 784613 421470 892777 927562 850407 897365 101996 534632 079502 502688 864052 818276 570728 899720 468770 479888 > 4²¹⁹ [i]