MinT - Best Known (23, 23+45, s)-Nets in Base 16

Best Known (23, 23+45, s)-Nets in Base 16

(23, 23+45, 65)-Net over F₁₆ — Constructive and digital

Digital (23, 68, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 68, 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+45, 104)-Net in Base 16 — Constructive

(23, 68, 104)-net in base 16, using

2 times m-reduction [i] based on (23, 70, 104)-net in base 16, using
- base change [i] based on digital (9, 56, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(23, 23+45, 129)-Net over F₁₆ — Digital

Digital (23, 68, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 68, 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+45, 2792)-Net in Base 16 — Upper bound on s

There is no (23, 68, 2793)-net in base 16, because

1 times m-reduction [i] would yield (23, 67, 2793)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 475 999256 131398 574963 156864 586393 040277 516434 842420 230836 751335 838396 166301 899416 > 16⁶⁷ [i]