MinT - Best Known (23, 23+33, s)-Nets in Base 16

Best Known (23, 23+33, s)-Nets in Base 16

(23, 23+33, 89)-Net over F₁₆ — Constructive and digital

Digital (23, 56, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, 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]

(23, 23+33, 120)-Net in Base 16 — Constructive

(23, 56, 120)-net in base 16, using

4 times m-reduction [i] based on (23, 60, 120)-net in base 16, using
- base change [i] based on digital (11, 48, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(23, 23+33, 129)-Net over F₁₆ — Digital

Digital (23, 56, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 56, 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+33, 6237)-Net in Base 16 — Upper bound on s

There is no (23, 56, 6238)-net in base 16, because

1 times m-reduction [i] would yield (23, 55, 6238)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 686938 345979 604336 505002 408453 860612 512995 833472 140037 427437 047871 > 16⁵⁵ [i]