MinT - Best Known (164−32, 164, s)-Nets in Base 4

Best Known (164−32, 164, s)-Nets in Base 4

(164−32, 164, 1094)-Net over F₄ — Constructive and digital

Digital (132, 164, 1094)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (20, 36, 66)-net over F₄, using
  - trace code for nets [i] based on digital (2, 18, 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 (96, 128, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 32, 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]

(164−32, 164, 6354)-Net over F₄ — Digital

Digital (132, 164, 6354)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−32, 164, 3361481)-Net in Base 4 — Upper bound on s

There is no (132, 164, 3361482)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 546 814032 317570 102681 407176 576585 113407 424807 773948 456159 479890 425013 277897 561843 837378 577911 906556 > 4¹⁶⁴ [i]