MinT - Best Known (240−61, 240, s)-Nets in Base 4

Best Known (240−61, 240, s)-Nets in Base 4

(240−61, 240, 548)-Net over F₄ — Constructive and digital

Digital (179, 240, 548)-net over F₄, using

4¹ times duplication [i] based on digital (178, 239, 548)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 35, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (143, 204, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 68, 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]

(240−61, 240, 648)-Net in Base 4 — Constructive

(179, 240, 648)-net in base 4, using

3 times m-reduction [i] based on (179, 243, 648)-net in base 4, using
- trace code for nets [i] based on (17, 81, 216)-net in base 64, using
  - 3 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]

(240−61, 240, 2009)-Net over F₄ — Digital

Digital (179, 240, 2009)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−61, 240, 251209)-Net in Base 4 — Upper bound on s

There is no (179, 240, 251210)-net in base 4, because

1 times m-reduction [i] would yield (179, 239, 251210)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 780469 324387 229361 065446 581718 982850 911862 773683 378967 140370 066819 158935 294886 502282 087199 936858 645585 677677 014695 996785 637461 068545 710614 474976 > 4²³⁹ [i]