MinT - Best Known (167−41, 167, s)-Nets in Base 4

Best Known (167−41, 167, s)-Nets in Base 4

(167−41, 167, 1028)-Net over F₄ — Constructive and digital

Digital (126, 167, 1028)-net over F₄, using

1 times m-reduction [i] based on digital (126, 168, 1028)-net over F₄, using
- trace code for nets [i] based on digital (0, 42, 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]

(167−41, 167, 1735)-Net over F₄ — Digital

Digital (126, 167, 1735)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−41, 167, 274952)-Net in Base 4 — Upper bound on s

There is no (126, 167, 274953)-net in base 4, because

1 times m-reduction [i] would yield (126, 166, 274953)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8749 443186 136644 719138 564994 844587 604714 971396 029657 514362 679823 053077 037353 874804 544503 033957 825856 > 4¹⁶⁶ [i]