MinT - Best Known (75−25, 75, s)-Nets in Base 7

Best Known (75−25, 75, s)-Nets in Base 7

(75−25, 75, 200)-Net over F₇ — Constructive and digital

Digital (50, 75, 200)-net over F₇, using

7¹ times duplication [i] based on digital (49, 74, 200)-net over F₇, using
- (u,Â u+v)-construction [i] based on
  1. 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]
  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 (see above)

(75−25, 75, 727)-Net over F₇ — Digital

Digital (50, 75, 727)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−25, 75, 143425)-Net in Base 7 — Upper bound on s

There is no (50, 75, 143426)-net in base 7, because

1 times m-reduction [i] would yield (50, 74, 143426)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 344 576129 017662 035908 840707 916411 771637 015783 735164 123006 694041 > 7⁷⁴ [i]