MinT - Best Known (27−16, 27, s)-Nets in Base 7

Best Known (27−16, 27, s)-Nets in Base 7

(27−16, 27, 20)-Net over F₇ — Constructive and digital

Digital (11, 27, 20)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 13)-net over F₇, using
  - 4 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (2, 18, 10)-net over F₇, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₇, using
    - Niederreiter sequence [i]
  - Niederreiter sequence (Bratleyâ€“Foxâ€“Niederreiter implementation) with equidistant coordinate [i]

(27−16, 27, 38)-Net over F₇ — Digital

Digital (11, 27, 38)-net over F₇, using

t-expansion [i] based on digital (9, 27, 38)-net over F₇, using
- net from sequence [i] based on digital (9, 37)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 9 and N(F) ≥ 38, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(27−16, 27, 441)-Net in Base 7 — Upper bound on s

There is no (11, 27, 442)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 66178 073694 406067 616145 > 7²⁷ [i]