MinT - Best Known (50, 50+29, s)-Nets in Base 16

Best Known (50, 50+29, s)-Nets in Base 16

(50, 50+29, 579)-Net over F₁₆ — Constructive and digital

Digital (50, 79, 579)-net over F₁₆, using

16¹ times duplication [i] based on digital (49, 78, 579)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 20, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [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]

(50, 50+29, 1894)-Net over F₁₆ — Digital

Digital (50, 79, 1894)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+29, 2060788)-Net in Base 16 — Upper bound on s

There is no (50, 79, 2060789)-net in base 16, because

1 times m-reduction [i] would yield (50, 78, 2060789)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8343 736708 369620 901653 690712 347967 603410 639832 989164 562515 408121 284617 714873 521353 318418 661816 > 16⁷⁸ [i]