MinT - Best Known (257−69, 257, s)-Nets in Base 4

Best Known (257−69, 257, s)-Nets in Base 4

(257−69, 257, 531)-Net over F₄ — Constructive and digital

Digital (188, 257, 531)-net over F₄, using

t-expansion [i] based on digital (179, 257, 531)-net over F₄, using
- 1 times m-reduction [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]

(257−69, 257, 576)-Net in Base 4 — Constructive

(188, 257, 576)-net in base 4, using

1 times m-reduction [i] based on (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]

(257−69, 257, 1666)-Net over F₄ — Digital

Digital (188, 257, 1666)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−69, 257, 153968)-Net in Base 4 — Upper bound on s

There is no (188, 257, 153969)-net in base 4, because

1 times m-reduction [i] would yield (188, 256, 153969)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13407 837768 689503 591993 720965 802157 431227 068851 682440 167278 203745 364790 921789 990121 925920 670166 800813 654607 094023 360059 011588 139880 229640 669380 574696 903976 > 4²⁵⁶ [i]