MinT - Best Known (62−19, 62, s)-Nets in Base 7

Best Known (62−19, 62, s)-Nets in Base 7

(62−19, 62, 208)-Net over F₇ — Constructive and digital

Digital (43, 62, 208)-net over F₇, using

generalized (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 (9, 18, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 9, 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 (19, 38, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 19, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(62−19, 62, 1034)-Net over F₇ — Digital

Digital (43, 62, 1034)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(62−19, 62, 369386)-Net in Base 7 — Upper bound on s

There is no (43, 62, 369387)-net in base 7, because

1 times m-reduction [i] would yield (43, 61, 369387)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 3556 238516 253510 567288 270879 842779 059911 443720 906387 > 7⁶¹ [i]