MinT - Best Known (63, 63+39, s)-Nets in Base 16

Best Known (63, 63+39, s)-Nets in Base 16

(63, 63+39, 563)-Net over F₁₆ — Constructive and digital

Digital (63, 102, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 24, 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 (39, 78, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 39, 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]

(63, 63+39, 1728)-Net over F₁₆ — Digital

Digital (63, 102, 1728)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(63, 63+39, 1330360)-Net in Base 16 — Upper bound on s

There is no (63, 102, 1330361)-net in base 16, because

1 times m-reduction [i] would yield (63, 101, 1330361)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 41 316504 602406 311858 945416 443229 534032 357574 414461 283065 946053 462028 594328 408180 825851 028314 455053 080757 915917 030924 725736 > 16¹⁰¹ [i]