MinT - Best Known (96−53, 96, s)-Nets in Base 7

Best Known (96−53, 96, s)-Nets in Base 7

(96−53, 96, 44)-Net over F₇ — Constructive and digital

Digital (43, 96, 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]

(96−53, 96, 105)-Net over F₇ — Digital

Digital (43, 96, 105)-net over F₇, using

net from sequence [i] based on digital (43, 104)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 43 and N(F) ≥ 105, using
  - an algebraic function field by Teo [i]

(96−53, 96, 2135)-Net in Base 7 — Upper bound on s

There is no (43, 96, 2136)-net in base 7, because

1 times m-reduction [i] would yield (43, 95, 2136)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 192 469708 534342 032204 426093 446320 504958 525346 694379 256992 949963 381717 333974 188369 > 7⁹⁵ [i]