MinT - Best Known (188−37, 188, s)-Nets in Base 4

Best Known (188−37, 188, s)-Nets in Base 4

(188−37, 188, 1094)-Net over F₄ — Constructive and digital

Digital (151, 188, 1094)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (22, 40, 66)-net over F₄, using
  - trace code for nets [i] based on digital (2, 20, 33)-net over F₁₆, using
    - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
2. digital (111, 148, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 37, 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]

(188−37, 188, 6651)-Net over F₄ — Digital

Digital (151, 188, 6651)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−37, 188, 4526283)-Net in Base 4 — Upper bound on s

There is no (151, 188, 4526284)-net in base 4, because

1 times m-reduction [i] would yield (151, 187, 4526284)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 38478 577603 995063 078196 834174 879665 386005 666958 476655 479896 014592 558563 911819 374712 084084 698574 005974 701772 170276 > 4¹⁸⁷ [i]