MinT - Best Known (172−44, 172, s)-Nets in Base 4

Best Known (172−44, 172, s)-Nets in Base 4

(172−44, 172, 531)-Net over F₄ — Constructive and digital

Digital (128, 172, 531)-net over F₄, using

t-expansion [i] based on digital (127, 172, 531)-net over F₄, using
- 8 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]

(172−44, 172, 576)-Net in Base 4 — Constructive

(128, 172, 576)-net in base 4, using

2 times m-reduction [i] based on (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]

(172−44, 172, 1462)-Net over F₄ — Digital

Digital (128, 172, 1462)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−44, 172, 153707)-Net in Base 4 — Upper bound on s

There is no (128, 172, 153708)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35 837012 122068 865928 645052 806346 561684 813821 327696 277532 408759 174167 374562 758095 545323 282945 429702 184652 > 4¹⁷² [i]