MinT - Best Known (251−68, 251, s)-Nets in Base 4

Best Known (251−68, 251, s)-Nets in Base 4

(251−68, 251, 531)-Net over F₄ — Constructive and digital

Digital (183, 251, 531)-net over F₄, using

t-expansion [i] based on digital (179, 251, 531)-net over F₄, using
- 7 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]

(251−68, 251, 576)-Net in Base 4 — Constructive

(183, 251, 576)-net in base 4, using

1 times m-reduction [i] based on (183, 252, 576)-net in base 4, using
- trace code for nets [i] based on (15, 84, 192)-net in base 64, using
  - base change [i] based on digital (3, 72, 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]

(251−68, 251, 1567)-Net over F₄ — Digital

Digital (183, 251, 1567)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−68, 251, 125567)-Net in Base 4 — Upper bound on s

There is no (183, 251, 125568)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 095306 281321 025809 102947 321780 173525 028157 285072 670038 713176 686270 183468 898938 569872 067026 181640 209835 936243 112706 719395 708287 089000 078929 801374 244005 > 4²⁵¹ [i]