MinT - Best Known (72−24, 72, s)-Nets in Base 7

Best Known (72−24, 72, s)-Nets in Base 7

(72−24, 72, 200)-Net over F₇ — Constructive and digital

Digital (48, 72, 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 (24, 48, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 24, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(72−24, 72, 706)-Net over F₇ — Digital

Digital (48, 72, 706)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−24, 72, 103697)-Net in Base 7 — Upper bound on s

There is no (48, 72, 103698)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 7 032334 303026 580473 749510 867090 050482 740862 087484 176501 499033 > 7⁷² [i]