MinT - Best Known (25, 25+51, s)-Nets in Base 16

Best Known (25, 25+51, s)-Nets in Base 16

(25, 25+51, 65)-Net over F₁₆ — Constructive and digital

Digital (25, 76, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 76, 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+51, 104)-Net in Base 16 — Constructive

(25, 76, 104)-net in base 16, using

4 times m-reduction [i] based on (25, 80, 104)-net in base 16, using
- base change [i] based on digital (9, 64, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(25, 25+51, 144)-Net over F₁₆ — Digital

Digital (25, 76, 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+51, 2765)-Net in Base 16 — Upper bound on s

There is no (25, 76, 2766)-net in base 16, because

1 times m-reduction [i] would yield (25, 75, 2766)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 048270 712718 530362 156687 625378 293223 438434 298561 245152 400499 651955 126739 775036 038212 432876 > 16⁷⁵ [i]