MinT - Best Known (174, 174+59, s)-Nets in Base 4

Best Known (174, 174+59, s)-Nets in Base 4

(174, 174+59, 548)-Net over F₄ — Constructive and digital

Digital (174, 233, 548)-net over F₄, using

4¹ times duplication [i] based on digital (173, 232, 548)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 34, 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 (139, 198, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 66, 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]

(174, 174+59, 648)-Net in Base 4 — Constructive

(174, 233, 648)-net in base 4, using

4² times duplication [i] based on (172, 231, 648)-net in base 4, using
- t-expansion [i] based on (170, 231, 648)-net in base 4, using
  - trace code for nets [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 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]

(174, 174+59, 1991)-Net over F₄ — Digital

Digital (174, 233, 1991)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(174, 174+59, 254941)-Net in Base 4 — Upper bound on s

There is no (174, 233, 254942)-net in base 4, because

1 times m-reduction [i] would yield (174, 232, 254942)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 637274 711154 986425 067347 845423 270813 530561 680484 777012 675398 643868 112871 177335 266425 494957 153568 927646 568383 374906 044452 383146 173049 468955 > 4²³² [i]