MinT - Best Known (185−69, 185, s)-Nets in Base 4

Best Known (185−69, 185, s)-Nets in Base 4

(185−69, 185, 139)-Net over F₄ — Constructive and digital

Digital (116, 185, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 35, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (81, 150, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 75, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(185−69, 185, 347)-Net over F₄ — Digital

Digital (116, 185, 347)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(185−69, 185, 8148)-Net in Base 4 — Upper bound on s

There is no (116, 185, 8149)-net in base 4, because

1 times m-reduction [i] would yield (116, 184, 8149)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 602 264738 127547 768940 151513 302142 106060 779121 970110 996870 435967 940036 257936 476532 134070 101234 797163 408633 650436 > 4¹⁸⁴ [i]