MinT - Best Known (34, 34+85, s)-Nets in Base 16

Best Known (34, 34+85, s)-Nets in Base 16

(34, 34+85, 65)-Net over F₁₆ — Constructive and digital

Digital (34, 119, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 119, 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]

(34, 34+85, 104)-Net in Base 16 — Constructive

(34, 119, 104)-net in base 16, using

6 times m-reduction [i] based on (34, 125, 104)-net in base 16, using
- base change [i] based on digital (9, 100, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(34, 34+85, 193)-Net over F₁₆ — Digital

Digital (34, 119, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 119, 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+85, 2635)-Net in Base 16 — Upper bound on s

There is no (34, 119, 2636)-net in base 16, because

1 times m-reduction [i] would yield (34, 118, 2636)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 12247 181806 697775 427972 260984 404339 761376 469380 300584 318445 289992 513862 221111 165223 926564 424220 508034 769464 363796 111196 881131 933912 621693 700906 > 16¹¹⁸ [i]