MinT - Best Known (40, 40+23, s)-Nets in Base 16

Best Known (40, 40+23, s)-Nets in Base 16

(40, 40+23, 579)-Net over F₁₆ — Constructive and digital

Digital (40, 63, 579)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 17, 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 (23, 46, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 23, 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+23, 1705)-Net over F₁₆ — Digital

Digital (40, 63, 1705)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 40+23, 2003455)-Net in Base 16 — Upper bound on s

There is no (40, 63, 2003456)-net in base 16, because

1 times m-reduction [i] would yield (40, 62, 2003456)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 452 313656 415358 193835 161360 488496 378429 856596 239483 254400 430527 556484 018241 > 16⁶² [i]