MinT - Best Known (89−34, 89, s)-Nets in Base 16

Best Known (89−34, 89, s)-Nets in Base 16

(89−34, 89, 559)-Net over F₁₆ — Constructive and digital

Digital (55, 89, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 21, 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 (34, 68, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 34, 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]

(89−34, 89, 1568)-Net over F₁₆ — Digital

Digital (55, 89, 1568)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−34, 89, 963323)-Net in Base 16 — Upper bound on s

There is no (55, 89, 963324)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146785 566215 740746 042743 449234 296093 242568 938023 655624 204993 934278 733467 249454 834453 119848 126285 716047 775121 > 16⁸⁹ [i]