MinT - Best Known (161−31, 161, s)-Nets in Base 4

Best Known (161−31, 161, s)-Nets in Base 4

(161−31, 161, 1104)-Net over F₄ — Constructive and digital

Digital (130, 161, 1104)-net over F₄, using

4¹ times duplication [i] based on digital (129, 160, 1104)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 36, 76)-net over F₄, using
    - trace code for nets [i] based on digital (3, 18, 38)-net over F₁₆, using
      - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
  2. digital (93, 124, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 31, 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]

(161−31, 161, 6849)-Net over F₄ — Digital

Digital (130, 161, 6849)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(161−31, 161, 5657410)-Net in Base 4 — Upper bound on s

There is no (130, 161, 5657411)-net in base 4, because

1 times m-reduction [i] would yield (130, 160, 5657411)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 135988 871059 183456 565890 889575 415953 689166 451088 688706 386777 801793 616700 424809 313275 527815 951360 > 4¹⁶⁰ [i]