MinT - Best Known (45, 45+19, s)-Nets in Base 7

Best Known (45, 45+19, s)-Nets in Base 7

(45, 45+19, 213)-Net over F₇ — Constructive and digital

Digital (45, 64, 213)-net over F₇, using

7¹ times duplication [i] based on digital (44, 63, 213)-net over F₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (1, 7, 13)-net over F₇, using
    - 6 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
      - a shift-net [i]
  2. digital (9, 18, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 9, 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 (19, 38, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 19, 50)-net over F₄₉, using
      - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(45, 45+19, 1282)-Net over F₇ — Digital

Digital (45, 64, 1282)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+19, 569222)-Net in Base 7 — Upper bound on s

There is no (45, 64, 569223)-net in base 7, because

1 times m-reduction [i] would yield (45, 63, 569223)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 174252 069808 316585 017303 078972 506048 067394 218814 452219 > 7⁶³ [i]