MinT - Best Known (88−55, 88, s)-Nets in Base 16

Best Known (88−55, 88, s)-Nets in Base 16

(88−55, 88, 82)-Net over F₁₆ — Constructive and digital

Digital (33, 88, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 27, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [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]

(88−55, 88, 120)-Net in Base 16 — Constructive

(33, 88, 120)-net in base 16, using

22 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]

(88−55, 88, 193)-Net over F₁₆ — Digital

Digital (33, 88, 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]

(88−55, 88, 5509)-Net in Base 16 — Upper bound on s

There is no (33, 88, 5510)-net in base 16, because

1 times m-reduction [i] would yield (33, 87, 5510)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 575 449378 505969 693786 361783 040788 876997 696569 561549 306375 798846 589086 768828 041164 302531 116787 710823 350176 > 16⁸⁷ [i]