MinT - Best Known (24, 57, s)-Nets in Base 16

Best Known (24, 57, s)-Nets in Base 16

(24, 57, 98)-Net over F₁₆ — Constructive and digital

Digital (24, 57, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 18, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (6, 39, 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, 57, 128)-Net in Base 16 — Constructive

(24, 57, 128)-net in base 16, using

base change [i] based on digital (5, 38, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(24, 57, 129)-Net over F₁₆ — Digital

Digital (24, 57, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 57, 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]

(24, 57, 133)-Net in Base 16

(24, 57, 133)-net in base 16, using

base change [i] based on digital (5, 38, 133)-net over F₆₄, using
- net from sequence [i] based on digital (5, 132)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 133, using
    - a curve from the manYPoints database [i]

(24, 57, 7419)-Net in Base 16 — Upper bound on s

There is no (24, 57, 7420)-net in base 16, because

1 times m-reduction [i] would yield (24, 56, 7420)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 26 997188 761334 589783 570944 238327 374541 573656 409758 474063 812672 938301 > 16⁵⁶ [i]