MinT - Best Known (68−20, 68, s)-Nets in Base 7

Best Known (68−20, 68, s)-Nets in Base 7

(68−20, 68, 213)-Net over F₇ — Constructive and digital

Digital (48, 68, 213)-net over F₇, using

7¹ times duplication [i] based on digital (47, 67, 213)-net over F₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (1, 7, 13)-net over F₇, using
    - 6 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
      - a shift-net [i]
  2. digital (10, 20, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 10, 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 (20, 40, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 20, 50)-net over F₄₉, using
      - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(68−20, 68, 1408)-Net over F₇ — Digital

Digital (48, 68, 1408)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−20, 68, 421197)-Net in Base 7 — Upper bound on s

There is no (48, 68, 421198)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 2928 707892 548611 106413 205979 642076 292569 880734 816660 698301 > 7⁶⁸ [i]