MinT - Best Known (172, 172+61, s)-Nets in Base 4

Best Known (172, 172+61, s)-Nets in Base 4

(172, 172+61, 531)-Net over F₄ — Constructive and digital

Digital (172, 233, 531)-net over F₄, using

t-expansion [i] based on digital (171, 233, 531)-net over F₄, using
- 13 times m-reduction [i] based on digital (171, 246, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 82, 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, 172+61, 648)-Net in Base 4 — Constructive

(172, 233, 648)-net in base 4, using

4² times duplication [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]

(172, 172+61, 1713)-Net over F₄ — Digital

Digital (172, 233, 1713)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+61, 181777)-Net in Base 4 — Upper bound on s

There is no (172, 233, 181778)-net in base 4, because

1 times m-reduction [i] would yield (172, 232, 181778)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 640137 406643 448556 252062 866103 279313 726212 314624 573450 187336 871201 081521 251475 259994 935826 172591 936467 269032 573923 239917 071005 485411 477248 > 4²³² [i]