MinT - Best Known (42, 42+25, s)-Nets in Base 16

Best Known (42, 42+25, s)-Nets in Base 16

(42, 42+25, 563)-Net over F₁₆ — Constructive and digital

Digital (42, 67, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 17, 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 (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]

(42, 42+25, 1514)-Net over F₁₆ — Digital

Digital (42, 67, 1514)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+25, 1478863)-Net in Base 16 — Upper bound on s

There is no (42, 67, 1478864)-net in base 16, because

1 times m-reduction [i] would yield (42, 66, 1478864)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 29 642797 705095 973193 350668 213559 487228 006617 275286 468357 915302 799867 330292 112271 > 16⁶⁶ [i]