MinT - Best Known (38−21, 38, s)-Nets in Base 7

Best Known (38−21, 38, s)-Nets in Base 7

(38−21, 38, 27)-Net over F₇ — Constructive and digital

Digital (17, 38, 27)-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 (6, 27, 14)-net over F₇, using
  - net from sequence [i] based on digital (6, 13)-sequence over F₇, using
    - Niederreiter sequence [i]

(38−21, 38, 52)-Net over F₇ — Digital

Digital (17, 38, 52)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(38−21, 38, 1004)-Net in Base 7 — Upper bound on s

There is no (17, 38, 1005)-net in base 7, because

1 times m-reduction [i] would yield (17, 37, 1005)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 18 617542 689987 041103 118348 829929 > 7³⁷ [i]