MinT - Best Known (84−57, 84, s)-Nets in Base 16

Best Known (84−57, 84, s)-Nets in Base 16

(84−57, 84, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 84, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 84, 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]

(84−57, 84, 104)-Net in Base 16 — Constructive

(27, 84, 104)-net in base 16, using

6 times m-reduction [i] based on (27, 90, 104)-net in base 16, using
- base change [i] based on digital (9, 72, 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]

(84−57, 84, 156)-Net over F₁₆ — Digital

Digital (27, 84, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(84−57, 84, 2778)-Net in Base 16 — Upper bound on s

There is no (27, 84, 2779)-net in base 16, because

1 times m-reduction [i] would yield (27, 83, 2779)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8755 014841 282576 022270 999644 835010 962960 656907 407719 276411 842893 287034 125937 478241 795905 983398 047056 > 16⁸³ [i]