MinT - Best Known (231−62, 231, s)-Nets in Base 4

Best Known (231−62, 231, s)-Nets in Base 4

(231−62, 231, 531)-Net over F₄ — Constructive and digital

Digital (169, 231, 531)-net over F₄, using

12 times m-reduction [i] based on digital (169, 243, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 81, 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]

(231−62, 231, 576)-Net in Base 4 — Constructive

(169, 231, 576)-net in base 4, using

t-expansion [i] based on (168, 231, 576)-net in base 4, using
- trace code for nets [i] based on (14, 77, 192)-net in base 64, using
  - base change [i] based on digital (3, 66, 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]

(231−62, 231, 1526)-Net over F₄ — Digital

Digital (169, 231, 1526)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−62, 231, 126806)-Net in Base 4 — Upper bound on s

There is no (169, 231, 126807)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 908755 272756 227610 895910 004846 012516 673094 849926 968165 650887 659458 453599 508181 749858 337890 919653 605710 435649 429106 743787 960061 089956 076000 > 4²³¹ [i]