MinT - Best Known (18, 18+21, s)-Nets in Base 7

Best Known (18, 18+21, s)-Nets in Base 7

(18, 18+21, 28)-Net over F₇ — Constructive and digital

Digital (18, 39, 28)-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 (7, 28, 15)-net over F₇, using
  - net from sequence [i] based on digital (7, 14)-sequence over F₇, using
    - Niederreiter sequence [i]

(18, 18+21, 58)-Net over F₇ — Digital

Digital (18, 39, 58)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(18, 18+21, 1221)-Net in Base 7 — Upper bound on s

There is no (18, 39, 1222)-net in base 7, because

1 times m-reduction [i] would yield (18, 38, 1222)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 130 102467 984408 129466 634222 533261 > 7³⁸ [i]