MinT - Best Known (76−29, 76, s)-Nets in Base 16

Best Known (76−29, 76, s)-Nets in Base 16

(76−29, 76, 559)-Net over F₁₆ — Constructive and digital

Digital (47, 76, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 18, 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 (29, 58, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 29, 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]

(76−29, 76, 1411)-Net over F₁₆ — Digital

Digital (47, 76, 1411)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(76−29, 76, 1137644)-Net in Base 16 — Upper bound on s

There is no (47, 76, 1137645)-net in base 16, because

1 times m-reduction [i] would yield (47, 75, 1137645)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 037059 318296 694951 342731 128672 731703 422519 524651 701740 346914 995932 428945 605688 136037 054576 > 16⁷⁵ [i]