MinT - Best Known (57, 87, s)-Nets in Base 4

Best Known (57, 87, s)-Nets in Base 4

(57, 87, 135)-Net over F₄ — Constructive and digital

Digital (57, 87, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 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 (42, 72, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 36, 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]

(57, 87, 248)-Net over F₄ — Digital

Digital (57, 87, 248)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 87, 6634)-Net in Base 4 — Upper bound on s

There is no (57, 87, 6635)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 23971 551407 960437 652177 996762 590283 727908 850885 022528 > 4⁸⁷ [i]