MinT - Best Known (34, 34+59, s)-Nets in Base 16

Best Known (34, 34+59, s)-Nets in Base 16

(34, 34+59, 71)-Net over F₁₆ — Constructive and digital

Digital (34, 93, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 31, 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, 62, 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]

(34, 34+59, 120)-Net in Base 16 — Constructive

(34, 93, 120)-net in base 16, using

22 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]

(34, 34+59, 193)-Net over F₁₆ — Digital

Digital (34, 93, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 93, 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]

(34, 34+59, 5124)-Net in Base 16 — Upper bound on s

There is no (34, 93, 5125)-net in base 16, because

1 times m-reduction [i] would yield (34, 92, 5125)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 602 968288 247362 031984 432987 872361 177702 150550 986909 742994 896203 618654 932807 949603 411407 007472 656030 740553 829376 > 16⁹² [i]