MinT - Best Known (40, 40+25, s)-Nets in Base 16

Best Known (40, 40+25, s)-Nets in Base 16

(40, 40+25, 552)-Net over F₁₆ — Constructive and digital

Digital (40, 65, 552)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (25, 50, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 25, 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]

(40, 40+25, 1204)-Net over F₁₆ — Digital

Digital (40, 65, 1204)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 40+25, 931623)-Net in Base 16 — Upper bound on s

There is no (40, 65, 931624)-net in base 16, because

1 times m-reduction [i] would yield (40, 64, 931624)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 115792 759431 911396 254919 476840 951041 942371 641087 880933 656643 719694 055944 439196 > 16⁶⁴ [i]