MinT - Best Known (71−47, 71, s)-Nets in Base 16

Best Known (71−47, 71, s)-Nets in Base 16

(71−47, 71, 65)-Net over F₁₆ — Constructive and digital

Digital (24, 71, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 71, 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]

(71−47, 71, 104)-Net in Base 16 — Constructive

(24, 71, 104)-net in base 16, using

4 times m-reduction [i] based on (24, 75, 104)-net in base 16, using
- base change [i] based on digital (9, 60, 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]

(71−47, 71, 129)-Net over F₁₆ — Digital

Digital (24, 71, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 71, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(71−47, 71, 2892)-Net in Base 16 — Upper bound on s

There is no (24, 71, 2893)-net in base 16, because

1 times m-reduction [i] would yield (24, 70, 2893)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 957726 862696 633364 920466 031206 457259 451568 569076 327280 373520 746888 945403 193188 839136 > 16⁷⁰ [i]