MinT - Best Known (23, 23+31, s)-Nets in Base 16

Best Known (23, 23+31, s)-Nets in Base 16

(23, 23+31, 98)-Net over F₁₆ — Constructive and digital

Digital (23, 54, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 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, 37, 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]

(23, 23+31, 128)-Net in Base 16 — Constructive

(23, 54, 128)-net in base 16, using

base change [i] based on digital (5, 36, 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]

(23, 23+31, 129)-Net over F₁₆ — Digital

Digital (23, 54, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 54, 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]

(23, 23+31, 133)-Net in Base 16

(23, 54, 133)-net in base 16, using

base change [i] based on digital (5, 36, 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]

(23, 23+31, 7687)-Net in Base 16 — Upper bound on s

There is no (23, 54, 7688)-net in base 16, because

1 times m-reduction [i] would yield (23, 53, 7688)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6589 361178 080810 450735 883215 051180 782210 610109 014256 095234 042176 > 16⁵³ [i]