MinT - Best Known (199−75, 199, s)-Nets in Base 4

Best Known (199−75, 199, s)-Nets in Base 4

(199−75, 199, 135)-Net over F₄ — Constructive and digital

Digital (124, 199, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 37, 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 (87, 162, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 81, 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]

(199−75, 199, 356)-Net over F₄ — Digital

Digital (124, 199, 356)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(199−75, 199, 8109)-Net in Base 4 — Upper bound on s

There is no (124, 199, 8110)-net in base 4, because

1 times m-reduction [i] would yield (124, 198, 8110)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 161593 955024 542385 013112 663036 989268 370802 344716 694360 492794 215033 540505 000488 842034 828227 635020 042635 936028 389970 006625 > 4¹⁹⁸ [i]