MinT - Best Known (30, 30+39, s)-Nets in Base 7

Best Known (30, 30+39, s)-Nets in Base 7

(30, 30+39, 33)-Net over F₇ — Constructive and digital

Digital (30, 69, 33)-net over F₇, using

t-expansion [i] based on digital (29, 69, 33)-net over F₇, using
- net from sequence [i] based on digital (29, 32)-sequence over F₇, using
  - Niederreiter sequence [i]

(30, 30+39, 78)-Net over F₇ — Digital

Digital (30, 69, 78)-net over F₇, using

t-expansion [i] based on digital (27, 69, 78)-net over F₇, using
- net from sequence [i] based on digital (27, 77)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 27 and N(F) ≥ 78, using
    - an algebraic function field by Teo [i]

(30, 30+39, 1386)-Net in Base 7 — Upper bound on s

There is no (30, 69, 1387)-net in base 7, because

1 times m-reduction [i] would yield (30, 68, 1387)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 2958 666791 157839 684634 476318 672289 116799 683427 570324 648079 > 7⁶⁸ [i]