MinT - Best Known (87−51, 87, s)-Nets in Base 16

Best Known (87−51, 87, s)-Nets in Base 16

(87−51, 87, 114)-Net over F₁₆ — Constructive and digital

Digital (36, 87, 114)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 30, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [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]

(87−51, 87, 177)-Net in Base 16 — Constructive

(36, 87, 177)-net in base 16, using

base change [i] based on digital (7, 58, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(87−51, 87, 193)-Net over F₁₆ — Digital

Digital (36, 87, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 87, 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]

(87−51, 87, 9398)-Net in Base 16 — Upper bound on s

There is no (36, 87, 9399)-net in base 16, because

1 times m-reduction [i] would yield (36, 86, 9399)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 35 846518 203626 251927 590373 474164 794299 264716 598307 283377 677772 358090 875146 783881 244339 734224 991168 618376 > 16⁸⁶ [i]