MinT - Best Known (81−50, 81, s)-Nets in Base 16

Best Known (81−50, 81, s)-Nets in Base 16

(81−50, 81, 82)-Net over F₁₆ — Constructive and digital

Digital (31, 81, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 25, 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, 56, 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]

(81−50, 81, 120)-Net in Base 16 — Constructive

(31, 81, 120)-net in base 16, using

19 times m-reduction [i] based on (31, 100, 120)-net in base 16, using
- base change [i] based on digital (11, 80, 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]

(81−50, 81, 168)-Net over F₁₆ — Digital

Digital (31, 81, 168)-net over F₁₆, using

net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
  - a curve from the manYPoints database [i]

(81−50, 81, 5392)-Net in Base 16 — Upper bound on s

There is no (31, 81, 5393)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 34 265581 674182 098457 960526 239909 691611 182096 829985 641105 784297 900173 915166 965776 104469 832279 877376 > 16⁸¹ [i]