MinT - Best Known (67−23, 67, s)-Nets in Base 7

Best Known (67−23, 67, s)-Nets in Base 7

(67−23, 67, 150)-Net over F₇ — Constructive and digital

Digital (44, 67, 150)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 21, 50)-net over F₇, using
  - base reduction for projective spaces (embedding PG(10,49) in PG(20,7)) for nets [i] based on digital (0, 11, 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 (23, 46, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 23, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(67−23, 67, 576)-Net over F₇ — Digital

Digital (44, 67, 576)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(67−23, 67, 96254)-Net in Base 7 — Upper bound on s

There is no (44, 67, 96255)-net in base 7, because

1 times m-reduction [i] would yield (44, 66, 96255)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 59 772778 411224 427676 251270 123667 167615 341265 924496 277439 > 7⁶⁶ [i]