MinT - Best Known (32−13, 32, s)-Nets in Base 7

Best Known (32−13, 32, s)-Nets in Base 7

(32−13, 32, 108)-Net over F₇ — Constructive and digital

Digital (19, 32, 108)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 8)-net over F₇, using
  - net from sequence [i] based on digital (0, 7)-sequence over F₇, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using
      - the rational function field F₇(x) [i]
    - Niederreiter sequence [i]
2. digital (13, 26, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 13, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
      - Niederreiter sequence [i]

(32−13, 32, 184)-Net over F₇ — Digital

Digital (19, 32, 184)-net over F₇, using

trace code for nets [i] based on digital (3, 16, 92)-net over F₄₉, using
- net from sequence [i] based on digital (3, 91)-sequence over F₄₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 3 and N(F) ≥ 92, using
    - a curve from the manYPoints database [i]

(32−13, 32, 11595)-Net in Base 7 — Upper bound on s

There is no (19, 32, 11596)-net in base 7, because

1 times m-reduction [i] would yield (19, 31, 11596)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 157 837624 685735 393033 445097 > 7³¹ [i]