MinT - Best Known (68, 103, s)-Nets in Base 7

Best Known (68, 103, s)-Nets in Base 7

(68, 103, 150)-Net over F₇ — Constructive and digital

Digital (68, 103, 150)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (16, 33, 50)-net over F₇, using
  - base reduction for projective spaces (embedding PG(16,49) in PG(32,7)) for nets [i] based on digital (0, 17, 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 (35, 70, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 35, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(68, 103, 836)-Net over F₇ — Digital

Digital (68, 103, 836)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 103, 140717)-Net in Base 7 — Upper bound on s

There is no (68, 103, 140718)-net in base 7, because

1 times m-reduction [i] would yield (68, 102, 140718)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 158 494870 447978 602048 525700 670133 274046 348631 356072 531764 715474 900388 398070 654227 909749 > 7¹⁰² [i]