MinT - Best Known (30−12, 30, s)-Nets in Base 7

Best Known (30−12, 30, s)-Nets in Base 7

(30−12, 30, 108)-Net over F₇ — Constructive and digital

Digital (18, 30, 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 (12, 24, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 12, 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]

(30−12, 30, 184)-Net over F₇ — Digital

Digital (18, 30, 184)-net over F₇, using

trace code for nets [i] based on digital (3, 15, 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]

(30−12, 30, 8382)-Net in Base 7 — Upper bound on s

There is no (18, 30, 8383)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 22 545473 027630 210980 221277 > 7³⁰ [i]