MinT - Best Known (86−55, 86, s)-Nets in Base 16

Best Known (86−55, 86, s)-Nets in Base 16

(86−55, 86, 66)-Net over F₁₆ — Constructive and digital

Digital (31, 86, 66)-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 (2, 57, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(86−55, 86, 120)-Net in Base 16 — Constructive

(31, 86, 120)-net in base 16, using

14 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]

(86−55, 86, 168)-Net over F₁₆ — Digital

Digital (31, 86, 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]

(86−55, 86, 4483)-Net in Base 16 — Upper bound on s

There is no (31, 86, 4484)-net in base 16, because

1 times m-reduction [i] would yield (31, 85, 4484)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 244190 569981 198441 747703 649586 938837 629617 470885 675976 535184 846204 065613 144965 639781 662051 809596 114896 > 16⁸⁵ [i]