MinT - Best Known (47−14, 47, s)-Nets in Base 7

Best Known (47−14, 47, s)-Nets in Base 7

(47−14, 47, 213)-Net over F₇ — Constructive and digital

Digital (33, 47, 213)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 5, 13)-net over F₇, using
  - 8 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (7, 14, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 7, 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]
3. digital (14, 28, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 14, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(47−14, 47, 1079)-Net over F₇ — Digital

Digital (33, 47, 1079)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(47−14, 47, 266066)-Net in Base 7 — Upper bound on s

There is no (33, 47, 266067)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 5243 399078 012832 059777 080088 284952 332855 > 7⁴⁷ [i]