MinT - Best Known (25, 25+49, s)-Nets in Base 16

Best Known (25, 25+49, s)-Nets in Base 16

(25, 25+49, 65)-Net over F₁₆ — Constructive and digital

Digital (25, 74, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 74, 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+49, 104)-Net in Base 16 — Constructive

(25, 74, 104)-net in base 16, using

6 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+49, 144)-Net over F₁₆ — Digital

Digital (25, 74, 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+49, 2991)-Net in Base 16 — Upper bound on s

There is no (25, 74, 2992)-net in base 16, because

1 times m-reduction [i] would yield (25, 73, 2992)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7980 839696 856840 334131 509963 626278 256167 578432 463364 291967 939114 258974 688369 129365 833046 > 16⁷³ [i]