MinT - Best Known (30, 30+13, s)-Nets in Base 7

Best Known (30, 30+13, s)-Nets in Base 7

(30, 30+13, 213)-Net over F₇ — Constructive and digital

Digital (30, 43, 213)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 5, 13)-net over F₇, using
  - 8 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (6, 12, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 6, 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 (13, 26, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 13, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(30, 30+13, 947)-Net over F₇ — Digital

Digital (30, 43, 947)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(30, 30+13, 410916)-Net in Base 7 — Upper bound on s

There is no (30, 43, 410917)-net in base 7, because

1 times m-reduction [i] would yield (30, 42, 410917)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 311976 479598 991410 181487 437079 198425 > 7⁴² [i]