MinT - Best Known (136−35, 136, s)-Nets in Base 4

Best Known (136−35, 136, s)-Nets in Base 4

(136−35, 136, 531)-Net over F₄ — Constructive and digital

Digital (101, 136, 531)-net over F₄, using

5 times m-reduction [i] based on digital (101, 141, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 47, 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]

(136−35, 136, 576)-Net in Base 4 — Constructive

(101, 136, 576)-net in base 4, using

4¹ times duplication [i] based on (100, 135, 576)-net in base 4, using
- trace code for nets [i] based on (10, 45, 192)-net in base 64, using
  - 4 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
    - base change [i] based on digital (3, 42, 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]

(136−35, 136, 1172)-Net over F₄ — Digital

Digital (101, 136, 1172)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−35, 136, 144493)-Net in Base 4 — Upper bound on s

There is no (101, 136, 144494)-net in base 4, because

1 times m-reduction [i] would yield (101, 135, 144494)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1897 308254 769733 489004 586323 201216 418074 444878 075474 093401 736083 469733 013961 537930 > 4¹³⁵ [i]