MinT - Best Known (84−25, 84, s)-Nets in Base 7

Best Known (84−25, 84, s)-Nets in Base 7

(84−25, 84, 213)-Net over F₇ — Constructive and digital

Digital (59, 84, 213)-net over F₇, using

7¹ times duplication [i] based on digital (58, 83, 213)-net over F₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (1, 9, 13)-net over F₇, using
    - 4 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
      - a shift-net [i]
  2. digital (12, 24, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 12, 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 (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 (see above)

(84−25, 84, 1495)-Net over F₇ — Digital

Digital (59, 84, 1495)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(84−25, 84, 617257)-Net in Base 7 — Upper bound on s

There is no (59, 84, 617258)-net in base 7, because

1 times m-reduction [i] would yield (59, 83, 617258)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 13904 074645 909252 314360 989511 116384 304560 079798 908972 719690 037924 623129 > 7⁸³ [i]