MinT - Best Known (67−41, 67, s)-Nets in Base 7

Best Known (67−41, 67, s)-Nets in Base 7

(67−41, 67, 31)-Net over F₇ — Constructive and digital

Digital (26, 67, 31)-net over F₇, using

t-expansion [i] based on digital (25, 67, 31)-net over F₇, using
- net from sequence [i] based on digital (25, 30)-sequence over F₇, using
  - Niederreiter sequence [i]

(67−41, 67, 72)-Net over F₇ — Digital

Digital (26, 67, 72)-net over F₇, using

t-expansion [i] based on digital (22, 67, 72)-net over F₇, using
- net from sequence [i] based on digital (22, 71)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 22 and N(F) ≥ 72, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(67−41, 67, 838)-Net in Base 7 — Upper bound on s

There is no (26, 67, 839)-net in base 7, because

1 times m-reduction [i] would yield (26, 66, 839)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 60 723359 387243 952037 778986 125496 515704 487074 994380 878729 > 7⁶⁶ [i]