MinT - Best Known (69, 69+43, s)-Nets in Base 16

Best Known (69, 69+43, s)-Nets in Base 16

(69, 69+43, 563)-Net over F₁₆ — Constructive and digital

Digital (69, 112, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 26, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (43, 86, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 43, 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]

(69, 69+43, 1810)-Net over F₁₆ — Digital

Digital (69, 112, 1810)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+43, 1339774)-Net in Base 16 — Upper bound on s

There is no (69, 112, 1339775)-net in base 16, because

1 times m-reduction [i] would yield (69, 111, 1339775)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 45 427558 214537 868077 605053 095699 190636 698630 233853 893141 171023 157938 409850 735471 851928 086427 881895 353753 393006 015225 864478 237705 479126 > 16¹¹¹ [i]