MinT - Best Known (86−51, 86, s)-Nets in Base 16

Best Known (86−51, 86, s)-Nets in Base 16

(86−51, 86, 110)-Net over F₁₆ — Constructive and digital

Digital (35, 86, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 29, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 57, 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]

(86−51, 86, 128)-Net in Base 16 — Constructive

(35, 86, 128)-net in base 16, using

4 times m-reduction [i] based on (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]

(86−51, 86, 193)-Net over F₁₆ — Digital

Digital (35, 86, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 86, 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]

(86−51, 86, 8410)-Net in Base 16 — Upper bound on s

There is no (35, 86, 8411)-net in base 16, because

1 times m-reduction [i] would yield (35, 85, 8411)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 241025 101163 374808 571084 270994 087905 890646 138580 974047 597280 591627 898605 613697 947266 523603 734373 184126 > 16⁸⁵ [i]