MinT - Best Known (49, 49+25, s)-Nets in Base 7

Best Known (49, 49+25, s)-Nets in Base 7

(49, 49+25, 200)-Net over F₇ — Constructive and digital

Digital (49, 74, 200)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (12, 24, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 12, 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 (25, 50, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 25, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(49, 49+25, 671)-Net over F₇ — Digital

Digital (49, 74, 671)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 49+25, 121954)-Net in Base 7 — Upper bound on s

There is no (49, 74, 121955)-net in base 7, because

1 times m-reduction [i] would yield (49, 73, 121955)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 49 225920 946433 457347 309797 324600 706871 231507 740744 846795 481049 > 7⁷³ [i]