MinT - Best Known (109−80, 109, s)-Nets in Base 16

Best Known (109−80, 109, s)-Nets in Base 16

(109−80, 109, 65)-Net over F₁₆ — Constructive and digital

Digital (29, 109, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 109, 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]

(109−80, 109, 98)-Net in Base 16 — Constructive

(29, 109, 98)-net in base 16, using

1 times m-reduction [i] based on (29, 110, 98)-net in base 16, using
- base change [i] based on digital (7, 88, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(109−80, 109, 161)-Net over F₁₆ — Digital

Digital (29, 109, 161)-net over F₁₆, using

net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(109−80, 109, 1986)-Net in Base 16 — Upper bound on s

There is no (29, 109, 1987)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 178698 177226 490109 009050 738938 945820 458161 139263 780225 702330 751827 514349 165420 571004 126295 406243 366138 376735 552496 357050 986112 795576 > 16¹⁰⁹ [i]