MinT - Best Known (75−47, 75, s)-Nets in Base 16

Best Known (75−47, 75, s)-Nets in Base 16

(75−47, 75, 71)-Net over F₁₆ — Constructive and digital

Digital (28, 75, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 25, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (3, 50, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(75−47, 75, 120)-Net in Base 16 — Constructive

(28, 75, 120)-net in base 16, using

10 times m-reduction [i] based on (28, 85, 120)-net in base 16, using
- base change [i] based on digital (11, 68, 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]

(75−47, 75, 156)-Net over F₁₆ — Digital

Digital (28, 75, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 75, 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]

(75−47, 75, 4691)-Net in Base 16 — Upper bound on s

There is no (28, 75, 4692)-net in base 16, because

1 times m-reduction [i] would yield (28, 74, 4692)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 127487 365088 485870 002395 179677 630664 337502 248877 415135 903446 992058 867290 900846 481162 228016 > 16⁷⁴ [i]