MinT - Best Known (72, 72+45, s)-Nets in Base 16

Best Known (72, 72+45, s)-Nets in Base 16

(72, 72+45, 563)-Net over F₁₆ — Constructive and digital

Digital (72, 117, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 27, 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 (45, 90, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 45, 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]

(72, 72+45, 1853)-Net over F₁₆ — Digital

Digital (72, 117, 1853)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+45, 1348199)-Net in Base 16 — Upper bound on s

There is no (72, 117, 1348200)-net in base 16, because

1 times m-reduction [i] would yield (72, 116, 1348200)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 47 634434 495360 170163 835824 018804 706296 428703 865084 470355 882564 139034 084939 528507 819702 978694 968487 612882 976368 587628 694520 427431 698134 769126 > 16¹¹⁶ [i]