MinT - Best Known (18, 18+23, s)-Nets in Base 7

Best Known (18, 18+23, s)-Nets in Base 7

(18, 18+23, 27)-Net over F₇ — Constructive and digital

Digital (18, 41, 27)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 13)-net over F₇, using
  - 1 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (6, 29, 14)-net over F₇, using
  - net from sequence [i] based on digital (6, 13)-sequence over F₇, using
    - Niederreiter sequence [i]

(18, 18+23, 51)-Net over F₇ — Digital

Digital (18, 41, 51)-net over F₇, using

net from sequence [i] based on digital (18, 50)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 18 and N(F) ≥ 51, using
  - an algebraic function field by Teo [i]

(18, 18+23, 961)-Net in Base 7 — Upper bound on s

There is no (18, 41, 962)-net in base 7, because

1 times m-reduction [i] would yield (18, 40, 962)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 6414 579299 118090 612480 784627 534009 > 7⁴⁰ [i]