MinT - Best Known (163, 163+60, s)-Nets in Base 4

Best Known (163, 163+60, s)-Nets in Base 4

(163, 163+60, 531)-Net over F₄ — Constructive and digital

Digital (163, 223, 531)-net over F₄, using

11 times m-reduction [i] based on digital (163, 234, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 78, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(163, 163+60, 576)-Net in Base 4 — Constructive

(163, 223, 576)-net in base 4, using

4¹ times duplication [i] based on (162, 222, 576)-net in base 4, using
- trace code for nets [i] based on (14, 74, 192)-net in base 64, using
  - 3 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
    - base change [i] based on digital (3, 66, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(163, 163+60, 1464)-Net over F₄ — Digital

Digital (163, 223, 1464)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(163, 163+60, 119919)-Net in Base 4 — Upper bound on s

There is no (163, 223, 119920)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 181 713632 141445 599994 425026 838115 363863 018358 560150 358906 015824 022101 843687 342994 559226 097018 979466 664658 531623 272344 448908 866816 162946 > 4²²³ [i]