MinT - Best Known (89−55, 89, s)-Nets in Base 16

Best Known (89−55, 89, s)-Nets in Base 16

(89−55, 89, 89)-Net over F₁₆ — Constructive and digital

Digital (34, 89, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 28, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [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]

(89−55, 89, 120)-Net in Base 16 — Constructive

(34, 89, 120)-net in base 16, using

26 times m-reduction [i] based on (34, 115, 120)-net in base 16, using
- base change [i] based on digital (11, 92, 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]

(89−55, 89, 193)-Net over F₁₆ — Digital

Digital (34, 89, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 89, 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]

(89−55, 89, 6106)-Net in Base 16 — Upper bound on s

There is no (34, 89, 6107)-net in base 16, because

1 times m-reduction [i] would yield (34, 88, 6107)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9187 569023 691511 226304 634865 995409 956782 554854 238305 901093 198532 766552 563135 949713 483205 641474 432439 497136 > 16⁸⁸ [i]