MinT - Best Known (163, 163+40, s)-Nets in Base 4

Best Known (163, 163+40, s)-Nets in Base 4

(163, 163+40, 1076)-Net over F₄ — Constructive and digital

Digital (163, 203, 1076)-net over F₄, using

4¹ times duplication [i] based on digital (162, 202, 1076)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (22, 42, 48)-net over F₄, using
    - trace code for nets [i] based on digital (1, 21, 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 (120, 160, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 40, 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]

(163, 163+40, 7003)-Net over F₄ — Digital

Digital (163, 203, 7003)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(163, 163+40, 3573484)-Net in Base 4 — Upper bound on s

There is no (163, 203, 3573485)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 165 264297 523262 008961 843901 812573 452788 894982 859949 728185 978699 349199 751256 153344 733651 614654 687451 879947 241944 788162 995914 > 4²⁰³ [i]