MinT - Best Known (85−53, 85, s)-Nets in Base 16

Best Known (85−53, 85, s)-Nets in Base 16

(85−53, 85, 82)-Net over F₁₆ — Constructive and digital

Digital (32, 85, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 26, 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, 59, 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]

(85−53, 85, 120)-Net in Base 16 — Constructive

(32, 85, 120)-net in base 16, using

20 times m-reduction [i] based on (32, 105, 120)-net in base 16, using
- base change [i] based on digital (11, 84, 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]

(85−53, 85, 168)-Net over F₁₆ — Digital

Digital (32, 85, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 85, 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]

(85−53, 85, 5448)-Net in Base 16 — Upper bound on s

There is no (32, 85, 5449)-net in base 16, because

1 times m-reduction [i] would yield (32, 84, 5449)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 140096 674663 178564 364450 908264 148540 610348 574269 814801 073410 861618 511010 073770 621165 446467 428403 640736 > 16⁸⁴ [i]