MinT - Best Known (24, 53, s)-Nets in Base 16

Best Known (24, 53, s)-Nets in Base 16

(24, 53, 110)-Net over F₁₆ — Constructive and digital

Digital (24, 53, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 18, 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, 35, 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]

(24, 53, 129)-Net in Base 16 — Constructive

(24, 53, 129)-net in base 16, using

3 times m-reduction [i] based on (24, 56, 129)-net in base 16, using
- base change [i] based on digital (0, 32, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]

(24, 53, 142)-Net over F₁₆ — Digital

Digital (24, 53, 142)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(24, 53, 161)-Net in Base 16

(24, 53, 161)-net in base 16, using

1 times m-reduction [i] based on (24, 54, 161)-net in base 16, using
- base change [i] based on digital (6, 36, 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]

(24, 53, 11954)-Net in Base 16 — Upper bound on s

There is no (24, 53, 11955)-net in base 16, because

1 times m-reduction [i] would yield (24, 52, 11955)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 411 473699 416007 449536 003606 693898 580845 545322 207882 174414 137676 > 16⁵² [i]