MinT - Best Known (181−47, 181, s)-Nets in Base 4

Best Known (181−47, 181, s)-Nets in Base 4

(181−47, 181, 531)-Net over F₄ — Constructive and digital

Digital (134, 181, 531)-net over F₄, using

t-expansion [i] based on digital (133, 181, 531)-net over F₄, using
- 8 times m-reduction [i] based on digital (133, 189, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 63, 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]

(181−47, 181, 576)-Net in Base 4 — Constructive

(134, 181, 576)-net in base 4, using

2 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]

(181−47, 181, 1426)-Net over F₄ — Digital

Digital (134, 181, 1426)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(181−47, 181, 161829)-Net in Base 4 — Upper bound on s

There is no (134, 181, 161830)-net in base 4, because

1 times m-reduction [i] would yield (134, 180, 161830)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 348574 148574 954443 043874 408669 956895 613350 080167 675563 778650 058534 499293 281298 825243 681066 177579 109752 415056 > 4¹⁸⁰ [i]