MinT - Best Known (71−25, 71, s)-Nets in Base 7

Best Known (71−25, 71, s)-Nets in Base 7

(71−25, 71, 126)-Net over F₇ — Constructive and digital

Digital (46, 71, 126)-net over F₇, using

7¹ times duplication [i] based on digital (45, 70, 126)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (8, 20, 26)-net over F₇, using
    - (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 (1, 13, 13)-net over F₇ (see above)
  2. digital (25, 50, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 25, 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]

(71−25, 71, 529)-Net over F₇ — Digital

Digital (46, 71, 529)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(71−25, 71, 74973)-Net in Base 7 — Upper bound on s

There is no (46, 71, 74974)-net in base 7, because

1 times m-reduction [i] would yield (46, 70, 74974)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 143524 986128 033590 769730 974827 572902 440688 729037 819395 193529 > 7⁷⁰ [i]