MinT - Best Known (38, 38+17, s)-Nets in Base 7

Best Known (38, 38+17, s)-Nets in Base 7

(38, 38+17, 208)-Net over F₇ — Constructive and digital

Digital (38, 55, 208)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 5, 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 (8, 16, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 8, 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 (17, 34, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 17, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(38, 38+17, 919)-Net over F₇ — Digital

Digital (38, 55, 919)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(38, 38+17, 317646)-Net in Base 7 — Upper bound on s

There is no (38, 55, 317647)-net in base 7, because

1 times m-reduction [i] would yield (38, 54, 317647)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 4318 176693 804142 736472 509228 108955 071628 325585 > 7⁵⁴ [i]