MinT - Best Known (61, 87, s)-Nets in Base 7

Best Known (61, 87, s)-Nets in Base 7

(61, 87, 213)-Net over F₇ — Constructive and digital

Digital (61, 87, 213)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 9, 13)-net over F₇, using
  - 4 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (13, 26, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 13, 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 (26, 52, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 26, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(61, 87, 1493)-Net over F₇ — Digital

Digital (61, 87, 1493)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(61, 87, 427467)-Net in Base 7 — Upper bound on s

There is no (61, 87, 427468)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 33 384180 458747 625296 480231 510375 958493 964828 052955 864124 363888 030183 589097 > 7⁸⁷ [i]