MinT - Best Known (34, 34+61, s)-Nets in Base 16

Best Known (34, 34+61, s)-Nets in Base 16

(34, 34+61, 66)-Net over F₁₆ — Constructive and digital

Digital (34, 95, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 32, 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 (2, 63, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(34, 34+61, 120)-Net in Base 16 — Constructive

(34, 95, 120)-net in base 16, using

20 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+61, 193)-Net over F₁₆ — Digital

Digital (34, 95, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 95, 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+61, 4743)-Net in Base 16 — Upper bound on s

There is no (34, 95, 4744)-net in base 16, because

1 times m-reduction [i] would yield (34, 94, 4744)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 154388 696913 266519 107020 731068 292756 687279 909567 467011 039559 092363 639753 060724 579718 534973 381742 302022 545746 684176 > 16⁹⁴ [i]