MinT - Best Known (81−42, 81, s)-Nets in Base 7

Best Known (81−42, 81, s)-Nets in Base 7

(81−42, 81, 40)-Net over F₇ — Constructive and digital

Digital (39, 81, 40)-net over F₇, using

net from sequence [i] based on digital (39, 39)-sequence over F₇, using
- base reduction for sequences [i] based on digital (0, 39)-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]

(81−42, 81, 111)-Net over F₇ — Digital

Digital (39, 81, 111)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−42, 81, 2616)-Net in Base 7 — Upper bound on s

There is no (39, 81, 2617)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 284 157394 846006 446479 025192 344210 851746 139632 508467 393715 337362 552951 > 7⁸¹ [i]