MinT - Best Known (65−22, 65, s)-Nets in Base 7

Best Known (65−22, 65, s)-Nets in Base 7

(65−22, 65, 150)-Net over F₇ — Constructive and digital

Digital (43, 65, 150)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 21, 50)-net over F₇, using
  - base reduction for projective spaces (embedding PG(10,49) in PG(20,7)) for nets [i] based on digital (0, 11, 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]
2. digital (22, 44, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 22, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(65−22, 65, 608)-Net over F₇ — Digital

Digital (43, 65, 608)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−22, 65, 80646)-Net in Base 7 — Upper bound on s

There is no (43, 65, 80647)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 8 538531 259772 846466 101343 289003 404344 629411 563423 307903 > 7⁶⁵ [i]