MinT - Best Known (247−64, 247, s)-Nets in Base 4

Best Known (247−64, 247, s)-Nets in Base 4

(247−64, 247, 541)-Net over F₄ — Constructive and digital

Digital (183, 247, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 34, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (149, 213, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 71, 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]

(247−64, 247, 648)-Net in Base 4 — Constructive

(183, 247, 648)-net in base 4, using

2 times m-reduction [i] based on (183, 249, 648)-net in base 4, using
- trace code for nets [i] based on (17, 83, 216)-net in base 64, using
  - 1 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 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]

(247−64, 247, 1889)-Net over F₄ — Digital

Digital (183, 247, 1889)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−64, 247, 189168)-Net in Base 4 — Upper bound on s

There is no (183, 247, 189169)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51148 099057 766982 681367 601872 011644 690681 201489 380076 770672 550881 504277 925037 521651 079041 738299 199259 535672 659740 261245 392140 735499 994009 864175 381565 > 4²⁴⁷ [i]