MinT - Best Known (67, 67+42, s)-Nets in Base 16

Best Known (67, 67+42, s)-Nets in Base 16

(67, 67+42, 559)-Net over F₁₆ — Constructive and digital

Digital (67, 109, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 25, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [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]

(67, 67+42, 1731)-Net over F₁₆ — Digital

Digital (67, 109, 1731)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+42, 1028851)-Net in Base 16 — Upper bound on s

There is no (67, 109, 1028852)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 177453 517035 129856 726114 408994 210850 778451 475104 398294 316525 501090 727408 875661 496097 559699 129728 587380 992460 023044 049278 067098 113256 > 16¹⁰⁹ [i]