MinT - Best Known (172, 172+63, s)-Nets in Base 4

Best Known (172, 172+63, s)-Nets in Base 4

(172, 172+63, 531)-Net over F₄ — Constructive and digital

Digital (172, 235, 531)-net over F₄, using

t-expansion [i] based on digital (171, 235, 531)-net over F₄, using
- 11 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+63, 576)-Net in Base 4 — Constructive

(172, 235, 576)-net in base 4, using

4⁴ times duplication [i] based on (168, 231, 576)-net in base 4, using
- trace code for nets [i] based on (14, 77, 192)-net in base 64, using
  - base change [i] based on digital (3, 66, 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, 172+63, 1556)-Net over F₄ — Digital

Digital (172, 235, 1556)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+63, 145016)-Net in Base 4 — Upper bound on s

There is no (172, 235, 145017)-net in base 4, because

1 times m-reduction [i] would yield (172, 234, 145017)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 762 210768 138911 332475 116476 420443 284941 045286 121998 276130 532045 722012 685035 192298 540101 103899 099535 609849 788083 322979 102623 680139 033336 053024 > 4²³⁴ [i]