MinT - Best Known (27−15, 27, s)-Nets in Base 7

Best Known (27−15, 27, s)-Nets in Base 7

(27−15, 27, 24)-Net over F₇ — Constructive and digital

Digital (12, 27, 24)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 13)-net over F₇, using
  - 5 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (4, 19, 12)-net over F₇, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₇, using
    - Niederreiter sequence [i]
  - Niederreiter sequence (Bratleyâ€“Foxâ€“Niederreiter implementation) with equidistant coordinate [i]

(27−15, 27, 42)-Net over F₇ — Digital

Digital (12, 27, 42)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(27−15, 27, 771)-Net in Base 7 — Upper bound on s

There is no (12, 27, 772)-net in base 7, because

1 times m-reduction [i] would yield (12, 26, 772)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 9425 024528 330167 359073 > 7²⁶ [i]