MinT - Best Known (71−51, 71, s)-Nets in Base 16

Best Known (71−51, 71, s)-Nets in Base 16

(71−51, 71, 65)-Net over F₁₆ — Constructive and digital

Digital (20, 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]

(71−51, 71, 76)-Net in Base 16 — Constructive

(20, 71, 76)-net in base 16, using

4 times m-reduction [i] based on (20, 75, 76)-net in base 16, using
- base change [i] based on digital (5, 60, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(71−51, 71, 129)-Net over F₁₆ — Digital

Digital (20, 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]

(71−51, 71, 1582)-Net in Base 16 — Upper bound on s

There is no (20, 71, 1583)-net in base 16, because

1 times m-reduction [i] would yield (20, 70, 1583)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 957902 959976 618421 348066 737548 045342 817610 071747 577075 143722 580533 528562 982725 304876 > 16⁷⁰ [i]