MinT - Best Known (40−21, 40, s)-Nets in Base 7

Best Known (40−21, 40, s)-Nets in Base 7

(40−21, 40, 29)-Net over F₇ — Constructive and digital

Digital (19, 40, 29)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 13)-net over F₇, using
  - 2 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (8, 29, 16)-net over F₇, using
  - net from sequence [i] based on digital (8, 15)-sequence over F₇, using
    - Niederreiter sequence [i]

(40−21, 40, 64)-Net over F₇ — Digital

Digital (19, 40, 64)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(40−21, 40, 1485)-Net in Base 7 — Upper bound on s

There is no (19, 40, 1486)-net in base 7, because

1 times m-reduction [i] would yield (19, 39, 1486)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 911 796119 307045 715527 874675 527613 > 7³⁹ [i]