MinT - Best Known (86−58, 86, s)-Nets in Base 16

Best Known (86−58, 86, s)-Nets in Base 16

(86−58, 86, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 86, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 86, 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]

(86−58, 86, 104)-Net in Base 16 — Constructive

(28, 86, 104)-net in base 16, using

9 times m-reduction [i] based on (28, 95, 104)-net in base 16, using
- base change [i] based on digital (9, 76, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(86−58, 86, 156)-Net over F₁₆ — Digital

Digital (28, 86, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 86, 156)-net over F₁₆, using
- net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
    - a curve from the manYPoints database [i]

(86−58, 86, 2880)-Net in Base 16 — Upper bound on s

There is no (28, 86, 2881)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 35 993867 809709 796055 644342 110995 226917 685793 486263 307629 972975 425822 465044 820701 753992 390514 802352 806336 > 16⁸⁶ [i]