MinT - Best Known (25, 36, s)-Nets in Base 7

Best Known (25, 36, s)-Nets in Base 7

(25, 36, 250)-Net over F₇ — Constructive and digital

Digital (25, 36, 250)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 4, 50)-net over F₇, using
  - net defined by OOA [i] based on linear OOA(7⁴, 50, F₇, 3, 3) (dual of [(50, 3), 146, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(7⁴, 50, F₇, 2, 3) (dual of [(50, 2), 96, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(7⁴, 50, F₇, 3) (dual of [50, 46, 4]-code or 50-cap in PG(3,7)), using
        ovoid in PG(3,Â 7) [i]
2. digital (5, 10, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 5, 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 (11, 22, 100)-net over F₇, using
  - trace code 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 (see above)

(25, 36, 837)-Net over F₇ — Digital

Digital (25, 36, 837)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(25, 36, 357575)-Net in Base 7 — Upper bound on s

There is no (25, 36, 357576)-net in base 7, because

1 times m-reduction [i] would yield (25, 35, 357576)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 378820 978232 667821 239452 459697 > 7³⁵ [i]