MinT - Best Known (258−70, 258, s)-Nets in Base 4

Best Known (258−70, 258, s)-Nets in Base 4

(258−70, 258, 531)-Net over F₄ — Constructive and digital

Digital (188, 258, 531)-net over F₄, using

t-expansion [i] based on digital (179, 258, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 86, 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]

(258−70, 258, 576)-Net in Base 4 — Constructive

(188, 258, 576)-net in base 4, using

trace code for nets [i] based on (16, 86, 192)-net in base 64, using
- 5 times m-reduction [i] based on (16, 91, 192)-net in base 64, using
  - base change [i] based on digital (3, 78, 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]

(258−70, 258, 1593)-Net over F₄ — Digital

Digital (188, 258, 1593)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−70, 258, 127082)-Net in Base 4 — Upper bound on s

There is no (188, 258, 127083)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 214536 124241 015766 987157 375931 082186 545309 363728 772765 714548 430127 124879 641725 850593 224088 325378 279302 471636 608961 697629 818286 183873 568804 914384 323877 699020 > 4²⁵⁸ [i]