MinT - Best Known (222−59, 222, s)-Nets in Base 4

Best Known (222−59, 222, s)-Nets in Base 4

(222−59, 222, 531)-Net over F₄ — Constructive and digital

Digital (163, 222, 531)-net over F₄, using

12 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]

(222−59, 222, 576)-Net in Base 4 — Constructive

(163, 222, 576)-net in base 4, using

t-expansion [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]

(222−59, 222, 1537)-Net over F₄ — Digital

Digital (163, 222, 1537)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−59, 222, 150676)-Net in Base 4 — Upper bound on s

There is no (163, 222, 150677)-net in base 4, because

1 times m-reduction [i] would yield (163, 221, 150677)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 358408 397788 343830 106250 216043 946651 864957 253571 603308 159807 123937 458891 713294 885177 152973 766783 207534 467208 718894 317056 037007 883840 > 4²²¹ [i]