MinT - Best Known (25, 25+39, s)-Nets in Base 16

Best Known (25, 25+39, s)-Nets in Base 16

(25, 25+39, 82)-Net over F₁₆ — Constructive and digital

Digital (25, 64, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (6, 45, 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]

(25, 25+39, 120)-Net in Base 16 — Constructive

(25, 64, 120)-net in base 16, using

6 times m-reduction [i] based on (25, 70, 120)-net in base 16, using
- base change [i] based on digital (11, 56, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(25, 25+39, 144)-Net over F₁₆ — Digital

Digital (25, 64, 144)-net over F₁₆, using

net from sequence [i] based on digital (25, 143)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 25 and N(F) ≥ 144, using
  - a curve from the manYPoints database [i]

(25, 25+39, 5186)-Net in Base 16 — Upper bound on s

There is no (25, 64, 5187)-net in base 16, because

1 times m-reduction [i] would yield (25, 63, 5187)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7250 300721 915611 923446 652832 162846 296135 954902 853463 358182 938215 462796 852096 > 16⁶³ [i]