MinT - Best Known (64, 64+27, s)-Nets in Base 7

Best Known (64, 64+27, s)-Nets in Base 7

(64, 64+27, 213)-Net over F₇ — Constructive and digital

Digital (64, 91, 213)-net over F₇, using

7¹ times duplication [i] based on digital (63, 90, 213)-net over F₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (1, 10, 13)-net over F₇, using
    - 3 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
      - a shift-net [i]
  2. digital (13, 26, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 13, 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 (27, 54, 100)-net over F₇, using
    - trace code for nets [i] based on digital (0, 27, 50)-net over F₄₉, using
      - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(64, 64+27, 1609)-Net over F₇ — Digital

Digital (64, 91, 1609)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+27, 669774)-Net in Base 7 — Upper bound on s

There is no (64, 91, 669775)-net in base 7, because

1 times m-reduction [i] would yield (64, 90, 669775)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 11450 670240 176045 569016 628024 726531 328331 241454 340566 352085 025934 135054 879251 > 7⁹⁰ [i]