MinT - Best Known (172, 172+81, s)-Nets in Base 4

Best Known (172, 172+81, s)-Nets in Base 4

(172, 172+81, 450)-Net over F₄ — Constructive and digital

Digital (172, 253, 450)-net over F₄, using

t-expansion [i] based on digital (170, 253, 450)-net over F₄, using
- 7 times m-reduction [i] based on digital (170, 260, 450)-net over F₄, using
  - trace code for nets [i] based on digital (40, 130, 225)-net over F₁₆, using
    - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
        a function field by GarcÃa and Quoos [i]

(172, 172+81, 786)-Net over F₄ — Digital

Digital (172, 253, 786)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+81, 32599)-Net in Base 4 — Upper bound on s

There is no (172, 253, 32600)-net in base 4, because

1 times m-reduction [i] would yield (172, 252, 32600)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 407048 004917 799239 462876 045738 716053 686260 131192 243013 461035 055766 669649 018462 361467 355156 311740 944975 507316 310142 784171 414913 868997 390896 748563 392988 > 4²⁵² [i]