MinT - Best Known (166, 225, s)-Nets in Base 4

Best Known (166, 225, s)-Nets in Base 4

(166, 225, 531)-Net over F₄ — Constructive and digital

Digital (166, 225, 531)-net over F₄, using

t-expansion [i] based on digital (165, 225, 531)-net over F₄, using
- 12 times m-reduction [i] based on digital (165, 237, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 79, 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]

(166, 225, 648)-Net in Base 4 — Constructive

(166, 225, 648)-net in base 4, using

trace code for nets [i] based on (16, 75, 216)-net in base 64, using
- 2 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
  - base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(166, 225, 1649)-Net over F₄ — Digital

Digital (166, 225, 1649)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166, 225, 173914)-Net in Base 4 — Upper bound on s

There is no (166, 225, 173915)-net in base 4, because

1 times m-reduction [i] would yield (166, 224, 173915)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 726 884553 137440 750812 956529 956571 888171 935530 453929 254597 897272 442963 108685 057720 023514 431725 418416 052518 834032 138079 216267 749317 525152 > 4²²⁴ [i]