MinT - Best Known (167−33, 167, s)-Nets in Base 4

Best Known (167−33, 167, s)-Nets in Base 4

(167−33, 167, 1076)-Net over F₄ — Constructive and digital

Digital (134, 167, 1076)-net over F₄, using

4¹ times duplication [i] based on digital (133, 166, 1076)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (18, 34, 48)-net over F₄, using
    - trace code for nets [i] based on digital (1, 17, 24)-net over F₁₆, using
      - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
  2. digital (99, 132, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 33, 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]

(167−33, 167, 5928)-Net over F₄ — Digital

Digital (134, 167, 5928)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−33, 167, 3997500)-Net in Base 4 — Upper bound on s

There is no (134, 167, 3997501)-net in base 4, because

1 times m-reduction [i] would yield (134, 166, 3997501)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8749 033639 067039 724121 771089 742709 999785 945720 351003 414779 574403 462393 605086 903886 791192 568251 477421 > 4¹⁶⁶ [i]