MinT - Best Known (66−24, 66, s)-Nets in Base 7

Best Known (66−24, 66, s)-Nets in Base 7

(66−24, 66, 120)-Net over F₇ — Constructive and digital

Digital (42, 66, 120)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 18, 20)-net over F₇, using
  - 2 times m-reduction [i] based on digital (6, 20, 20)-net over F₇, using
    - a shift-net [i]
2. digital (24, 48, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 24, 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]

(66−24, 66, 430)-Net over F₇ — Digital

Digital (42, 66, 430)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(66−24, 66, 39189)-Net in Base 7 — Upper bound on s

There is no (42, 66, 39190)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 59 783096 002656 478624 410180 562375 728187 478751 073567 271353 > 7⁶⁶ [i]