MinT - Best Known (90−55, 90, s)-Nets in Base 16

Best Known (90−55, 90, s)-Nets in Base 16

(90−55, 90, 98)-Net over F₁₆ — Constructive and digital

Digital (35, 90, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 29, 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 (6, 61, 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]

(90−55, 90, 128)-Net in Base 16 — Constructive

(35, 90, 128)-net in base 16, using

base change [i] based on digital (5, 60, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(90−55, 90, 193)-Net over F₁₆ — Digital

Digital (35, 90, 193)-net over F₁₆, using

t-expansion [i] based on 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]

(90−55, 90, 6768)-Net in Base 16 — Upper bound on s

There is no (35, 90, 6769)-net in base 16, because

1 times m-reduction [i] would yield (35, 89, 6769)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146973 516015 546300 032715 906735 673240 111260 934016 808822 419226 503090 819411 161320 296543 773705 725772 699599 721696 > 16⁸⁹ [i]