MinT - Best Known (108−65, 108, s)-Nets in Base 7

Best Known (108−65, 108, s)-Nets in Base 7

(108−65, 108, 44)-Net over F₇ — Constructive and digital

Digital (43, 108, 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]

(108−65, 108, 105)-Net over F₇ — Digital

Digital (43, 108, 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]

(108−65, 108, 1406)-Net in Base 7 — Upper bound on s

There is no (43, 108, 1407)-net in base 7, because

1 times m-reduction [i] would yield (43, 107, 1407)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 2 687860 673871 463476 769329 097467 252333 987132 334390 845410 782197 950522 253975 804835 353781 690177 > 7¹⁰⁷ [i]