MinT - Best Known (21, 21+23, s)-Nets in Base 16

Best Known (21, 21+23, s)-Nets in Base 16

(21, 21+23, 110)-Net over F₁₆ — Constructive and digital

Digital (21, 44, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 15, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 29, 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]

(21, 21+23, 150)-Net in Base 16 — Constructive

(21, 44, 150)-net in base 16, using

1 times m-reduction [i] based on (21, 45, 150)-net in base 16, using
- base change [i] based on (6, 30, 150)-net in base 64, using
  - 5 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
    - base change [i] based on digital (1, 30, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(21, 21+23, 160)-Net over F₁₆ — Digital

Digital (21, 44, 160)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(21, 21+23, 161)-Net in Base 16

(21, 44, 161)-net in base 16, using

1 times m-reduction [i] based on (21, 45, 161)-net in base 16, using
- base change [i] based on digital (6, 30, 161)-net over F₆₄, using
  - net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(21, 21+23, 16664)-Net in Base 16 — Upper bound on s

There is no (21, 44, 16665)-net in base 16, because

1 times m-reduction [i] would yield (21, 43, 16665)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 5988 936776 644431 145692 331892 378435 423992 540349 940976 > 16⁴³ [i]