MinT - Best Known (90−47, 90, s)-Nets in Base 7

Best Known (90−47, 90, s)-Nets in Base 7

(90−47, 90, 44)-Net over F₇ — Constructive and digital

Digital (43, 90, 44)-net over F₇, using

net from sequence [i] based on digital (43, 43)-sequence over F₇, using
- base reduction for sequences [i] based on digital (0, 43)-sequence over F₄₉, using
  - s-reduction 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]

(90−47, 90, 117)-Net over F₇ — Digital

Digital (43, 90, 117)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−47, 90, 2912)-Net in Base 7 — Upper bound on s

There is no (43, 90, 2913)-net in base 7, because

1 times m-reduction [i] would yield (43, 89, 2913)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1642 357215 639428 138379 158910 568209 797468 331575 853258 902777 719665 949897 991287 > 7⁸⁹ [i]