MinT - Best Known (33, 33+53, s)-Nets in Base 16

Best Known (33, 33+53, s)-Nets in Base 16

(33, 33+53, 89)-Net over F₁₆ — Constructive and digital

Digital (33, 86, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 27, 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, 59, 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]

(33, 33+53, 120)-Net in Base 16 — Constructive

(33, 86, 120)-net in base 16, using

24 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]

(33, 33+53, 193)-Net over F₁₆ — Digital

Digital (33, 86, 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]

(33, 33+53, 6063)-Net in Base 16 — Upper bound on s

There is no (33, 86, 6064)-net in base 16, because

1 times m-reduction [i] would yield (33, 85, 6064)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 243621 666337 915026 128905 060615 136654 586205 782445 991828 550740 362601 385999 188790 384017 025472 363526 418086 > 16⁸⁵ [i]