MinT - Best Known (174−46, 174, s)-Nets in Base 4

Best Known (174−46, 174, s)-Nets in Base 4

(174−46, 174, 531)-Net over F₄ — Constructive and digital

Digital (128, 174, 531)-net over F₄, using

t-expansion [i] based on digital (127, 174, 531)-net over F₄, using
- 6 times m-reduction [i] based on digital (127, 180, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 60, 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]

(174−46, 174, 576)-Net in Base 4 — Constructive

(128, 174, 576)-net in base 4, using

trace code for nets [i] based on (12, 58, 192)-net in base 64, using
- 5 times m-reduction [i] based on (12, 63, 192)-net in base 64, using
  - base change [i] based on digital (3, 54, 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]

(174−46, 174, 1273)-Net over F₄ — Digital

Digital (128, 174, 1273)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(174−46, 174, 112713)-Net in Base 4 — Upper bound on s

There is no (128, 174, 112714)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 573 375683 066433 062040 984868 327680 161064 065701 731564 146946 442636 831424 695586 339975 336664 770784 105682 416592 > 4¹⁷⁴ [i]