MinT - Best Known (90−57, 90, s)-Nets in Base 16

Best Known (90−57, 90, s)-Nets in Base 16

(90−57, 90, 71)-Net over F₁₆ — Constructive and digital

Digital (33, 90, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 30, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (3, 60, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(90−57, 90, 120)-Net in Base 16 — Constructive

(33, 90, 120)-net in base 16, using

20 times m-reduction [i] based on (33, 110, 120)-net in base 16, using
- base change [i] based on digital (11, 88, 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]

(90−57, 90, 193)-Net over F₁₆ — Digital

Digital (33, 90, 193)-net over F₁₆, using

net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
  - a curve from the manYPoints database [i]

(90−57, 90, 5046)-Net in Base 16 — Upper bound on s

There is no (33, 90, 5047)-net in base 16, because

1 times m-reduction [i] would yield (33, 89, 5047)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 147328 565738 054019 820039 481868 368584 369566 429567 857235 135656 362428 841497 002326 919269 896591 372276 293576 371516 > 16⁸⁹ [i]