MinT - Best Known (44−23, 44, s)-Nets in Base 7

Best Known (44−23, 44, s)-Nets in Base 7

(44−23, 44, 30)-Net over F₇ — Constructive and digital

Digital (21, 44, 30)-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 (9, 32, 17)-net over F₇, using
  - net from sequence [i] based on digital (9, 16)-sequence over F₇, using
    - Niederreiter sequence [i]

(44−23, 44, 70)-Net over F₇ — Digital

Digital (21, 44, 70)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(44−23, 44, 1639)-Net in Base 7 — Upper bound on s

There is no (21, 44, 1640)-net in base 7, because

1 times m-reduction [i] would yield (21, 43, 1640)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 2 195970 866511 888742 870637 828256 410113 > 7⁴³ [i]