MinT - Best Known (78−53, 78, s)-Nets in Base 7

Best Known (78−53, 78, s)-Nets in Base 7

(78−53, 78, 31)-Net over F₇ — Constructive and digital

Digital (25, 78, 31)-net over F₇, using

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

(78−53, 78, 72)-Net over F₇ — Digital

Digital (25, 78, 72)-net over F₇, using

t-expansion [i] based on digital (22, 78, 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]

(78−53, 78, 542)-Net in Base 7 — Upper bound on s

There is no (25, 78, 543)-net in base 7, because

1 times m-reduction [i] would yield (25, 77, 543)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 118193 036020 007941 444642 768490 513322 779627 101889 964069 720339 584677 > 7⁷⁷ [i]