MinT - Best Known (13, 13+17, s)-Nets in Base 16

Best Known (13, 13+17, s)-Nets in Base 16

(13, 13+17, 71)-Net over F₁₆ — Constructive and digital

Digital (13, 30, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 10, 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 (3, 20, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(13, 13+17, 97)-Net over F₁₆ — Digital

Digital (13, 30, 97)-net over F₁₆, using

net from sequence [i] based on digital (13, 96)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 13 and N(F) ≥ 97, using
  - a curve from the manYPoints database [i]

(13, 13+17, 129)-Net in Base 16 — Constructive

(13, 30, 129)-net in base 16, using

base change [i] based on (3, 20, 129)-net in base 64, using
- 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
  - base change [i] based on digital (0, 18, 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]

(13, 13+17, 5810)-Net in Base 16 — Upper bound on s

There is no (13, 30, 5811)-net in base 16, because

1 times m-reduction [i] would yield (13, 29, 5811)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 83104 921443 660261 002354 463525 992746 > 16²⁹ [i]