MinT - Best Known (66, 66+28, s)-Nets in Base 7

Best Known (66, 66+28, s)-Nets in Base 7

(66, 66+28, 213)-Net over F₇ — Constructive and digital

Digital (66, 94, 213)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 10, 13)-net over F₇, using
  - 3 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. 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, 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 (28, 56, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 28, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(66, 66+28, 1607)-Net over F₇ — Digital

Digital (66, 94, 1607)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(66, 66+28, 475913)-Net in Base 7 — Upper bound on s

There is no (66, 94, 475914)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 27 492911 408488 717656 601833 940756 106109 201639 207957 830619 546619 539254 280984 231181 > 7⁹⁴ [i]