MinT - Best Known (27, 27+43, s)-Nets in Base 16

Best Known (27, 27+43, s)-Nets in Base 16

(27, 27+43, 82)-Net over F₁₆ — Constructive and digital

Digital (27, 70, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 21, 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, 49, 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]

(27, 27+43, 120)-Net in Base 16 — Constructive

(27, 70, 120)-net in base 16, using

10 times m-reduction [i] based on (27, 80, 120)-net in base 16, using
- base change [i] based on digital (11, 64, 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]

(27, 27+43, 156)-Net over F₁₆ — Digital

Digital (27, 70, 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]

(27, 27+43, 5222)-Net in Base 16 — Upper bound on s

There is no (27, 70, 5223)-net in base 16, because

1 times m-reduction [i] would yield (27, 69, 5223)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 121833 442323 281171 123163 302219 429604 323294 513874 455953 669886 266469 599669 408242 604996 > 16⁶⁹ [i]