MinT - Best Known (52, 52+27, s)-Nets in Base 7

Best Known (52, 52+27, s)-Nets in Base 7

(52, 52+27, 150)-Net over F₇ — Constructive and digital

Digital (52, 79, 150)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (12, 25, 50)-net over F₇, using
  - base reduction for projective spaces (embedding PG(12,49) in PG(24,7)) 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, 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 (27, 54, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 27, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(52, 52+27, 663)-Net over F₇ — Digital

Digital (52, 79, 663)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+27, 111124)-Net in Base 7 — Upper bound on s

There is no (52, 79, 111125)-net in base 7, because

1 times m-reduction [i] would yield (52, 78, 111125)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 827282 602703 194981 563046 735339 448897 756041 645899 424139 326094 994751 > 7⁷⁸ [i]