MinT - Best Known (68, 68+42, s)-Nets in Base 16

Best Known (68, 68+42, s)-Nets in Base 16

(68, 68+42, 563)-Net over F₁₆ — Constructive and digital

Digital (68, 110, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 26, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (42, 84, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 42, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(68, 68+42, 1850)-Net over F₁₆ — Digital

Digital (68, 110, 1850)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+42, 1174065)-Net in Base 16 — Upper bound on s

There is no (68, 110, 1174066)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 839251 234337 423240 047064 623504 216198 721696 083338 109667 814368 644758 774580 979817 920110 779069 961429 070828 802006 824817 613234 686022 167416 > 16¹¹⁰ [i]