MinT - Best Known (85, 85+49, s)-Nets in Base 4

Best Known (85, 85+49, s)-Nets in Base 4

(85, 85+49, 135)-Net over F₄ — Constructive and digital

Digital (85, 134, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 24, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (61, 110, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 55, 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]

(85, 85+49, 282)-Net over F₄ — Digital

Digital (85, 134, 282)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+49, 7070)-Net in Base 4 — Upper bound on s

There is no (85, 134, 7071)-net in base 4, because

1 times m-reduction [i] would yield (85, 133, 7071)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 118 691265 665528 402069 458631 677771 212336 072929 059045 036338 780160 942779 728719 220791 > 4¹³³ [i]