MinT - Best Known (85−33, 85, s)-Nets in Base 16

Best Known (85−33, 85, s)-Nets in Base 16

(85−33, 85, 552)-Net over F₁₆ — Constructive and digital

Digital (52, 85, 552)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 19, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (33, 66, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 33, 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]

(85−33, 85, 1363)-Net over F₁₆ — Digital

Digital (52, 85, 1363)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−33, 85, 950765)-Net in Base 16 — Upper bound on s

There is no (52, 85, 950766)-net in base 16, because

1 times m-reduction [i] would yield (52, 84, 950766)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 139984 981255 367499 141413 227897 092869 548837 580394 008230 726777 660916 376541 363096 216750 395332 643166 798966 > 16⁸⁴ [i]