MinT - Best Known (174, 234, s)-Nets in Base 4

Best Known (174, 234, s)-Nets in Base 4

(174, 234, 545)-Net over F₄ — Constructive and digital

Digital (174, 234, 545)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 33, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (141, 201, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 67, 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, 234, 648)-Net in Base 4 — Constructive

(174, 234, 648)-net in base 4, using

4³ times duplication [i] based on (171, 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, 234, 1887)-Net over F₄ — Digital

Digital (174, 234, 1887)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(174, 234, 199380)-Net in Base 4 — Upper bound on s

There is no (174, 234, 199381)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 762 239545 341965 026489 601903 688520 833227 518481 412068 536248 779121 767756 084082 453575 839879 058453 686584 415747 508124 750248 920989 361175 393607 436768 > 4²³⁴ [i]