MinT - Best Known (188−48, 188, s)-Nets in Base 4

Best Known (188−48, 188, s)-Nets in Base 4

(188−48, 188, 531)-Net over F₄ — Constructive and digital

Digital (140, 188, 531)-net over F₄, using

t-expansion [i] based on digital (139, 188, 531)-net over F₄, using
- 10 times m-reduction [i] based on digital (139, 198, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 66, 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]

(188−48, 188, 648)-Net in Base 4 — Constructive

(140, 188, 648)-net in base 4, using

1 times m-reduction [i] based on (140, 189, 648)-net in base 4, using
- trace code for nets [i] based on (14, 63, 216)-net in base 64, using
  - base change [i] based on digital (5, 54, 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]

(188−48, 188, 1591)-Net over F₄ — Digital

Digital (140, 188, 1591)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−48, 188, 169952)-Net in Base 4 — Upper bound on s

There is no (140, 188, 169953)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 153919 017479 096441 489593 087469 693311 922176 431070 051204 248470 442943 075071 416652 611234 470784 471029 218658 410762 146356 > 4¹⁸⁸ [i]