MinT - Best Known (67, 96, s)-Nets in Base 7

Best Known (67, 96, s)-Nets in Base 7

(67, 96, 213)-Net over F₇ — Constructive and digital

Digital (67, 96, 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 (29, 58, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 29, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(67, 96, 1501)-Net over F₇ — Digital

Digital (67, 96, 1501)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 96, 546881)-Net in Base 7 — Upper bound on s

There is no (67, 96, 546882)-net in base 7, because

1 times m-reduction [i] would yield (67, 95, 546882)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 192 450112 537516 377440 315310 759711 713155 889051 670606 667794 973724 172903 560335 303069 > 7⁹⁵ [i]