MinT - Best Known (34, 34+65, s)-Nets in Base 16

Best Known (34, 34+65, s)-Nets in Base 16

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

Digital (34, 99, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 99, 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+65, 120)-Net in Base 16 — Constructive

(34, 99, 120)-net in base 16, using

16 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+65, 193)-Net over F₁₆ — Digital

Digital (34, 99, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 99, 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+65, 4135)-Net in Base 16 — Upper bound on s

There is no (34, 99, 4136)-net in base 16, because

1 times m-reduction [i] would yield (34, 98, 4136)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 10101 523213 461687 408012 170848 021236 550532 860039 664485 045937 773345 756591 855950 087869 103139 958123 586423 935623 203297 815881 > 16⁹⁸ [i]