MinT - Best Known (136, 136+49, s)-Nets in Base 4

Best Known (136, 136+49, s)-Nets in Base 4

(136, 136+49, 531)-Net over F₄ — Constructive and digital

Digital (136, 185, 531)-net over F₄, using

t-expansion [i] based on digital (135, 185, 531)-net over F₄, using
- 7 times m-reduction [i] based on digital (135, 192, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 64, 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+49, 576)-Net in Base 4 — Constructive

(136, 185, 576)-net in base 4, using

1 times m-reduction [i] based on (136, 186, 576)-net in base 4, using
- trace code for nets [i] based on (12, 62, 192)-net in base 64, using
  - 1 times m-reduction [i] based on (12, 63, 192)-net in base 64, using
    - base change [i] based on digital (3, 54, 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, 136+49, 1330)-Net over F₄ — Digital

Digital (136, 185, 1330)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 136+49, 134887)-Net in Base 4 — Upper bound on s

There is no (136, 185, 134888)-net in base 4, because

1 times m-reduction [i] would yield (136, 184, 134888)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 601 271506 579036 869684 070149 135456 578423 937367 122285 084871 792441 205613 044107 680852 182466 942573 400437 399592 770556 > 4¹⁸⁴ [i]