MinT - Best Known (182−47, 182, s)-Nets in Base 4

Best Known (182−47, 182, s)-Nets in Base 4

(182−47, 182, 531)-Net over F₄ — Constructive and digital

Digital (135, 182, 531)-net over F₄, using

10 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]

(182−47, 182, 576)-Net in Base 4 — Constructive

(135, 182, 576)-net in base 4, using

t-expansion [i] based on (134, 182, 576)-net in base 4, using
- 1 times m-reduction [i] based on (134, 183, 576)-net in base 4, using
  - trace code for nets [i] based on (12, 61, 192)-net in base 64, using
    - 2 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]

(182−47, 182, 1469)-Net over F₄ — Digital

Digital (135, 182, 1469)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(182−47, 182, 171885)-Net in Base 4 — Upper bound on s

There is no (135, 182, 171886)-net in base 4, because

1 times m-reduction [i] would yield (135, 181, 171886)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 395301 376048 723527 245606 386798 757742 932151 883507 591234 728642 605889 338223 210001 218574 747854 728337 724691 951704 > 4¹⁸¹ [i]