MinT - Best Known (136, 136+46, s)-Nets in Base 4

Best Known (136, 136+46, s)-Nets in Base 4

(136, 136+46, 536)-Net over F₄ — Constructive and digital

Digital (136, 182, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 23, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (113, 159, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 53, 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, 136+46, 648)-Net in Base 4 — Constructive

(136, 182, 648)-net in base 4, using

1 times m-reduction [i] based on (136, 183, 648)-net in base 4, using
- trace code for nets [i] based on (14, 61, 216)-net in base 64, using
  - 2 times m-reduction [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]

(136, 136+46, 1622)-Net over F₄ — Digital

Digital (136, 182, 1622)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+46, 182564)-Net in Base 4 — Upper bound on s

There is no (136, 182, 182565)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 37 576701 131485 468208 836589 580799 285396 581534 856400 379957 307410 561966 355332 843751 485653 553995 664101 321195 799136 > 4¹⁸² [i]