MinT - Best Known (77−27, 77, s)-Nets in Base 7

Best Known (77−27, 77, s)-Nets in Base 7

(77−27, 77, 124)-Net over F₇ — Constructive and digital

Digital (50, 77, 124)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 19, 20)-net over F₇, using
  - 1 times m-reduction [i] based on digital (6, 20, 20)-net over F₇, using
    - a shift-net [i]
2. digital (31, 58, 104)-net over F₇, using
  - trace code for nets [i] based on digital (2, 29, 52)-net over F₄₉, using
    - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(77−27, 77, 573)-Net over F₇ — Digital

Digital (50, 77, 573)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−27, 77, 82373)-Net in Base 7 — Upper bound on s

There is no (50, 77, 82374)-net in base 7, because

1 times m-reduction [i] would yield (50, 76, 82374)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 16885 644728 258160 513135 997743 052617 867684 004002 068632 047778 916173 > 7⁷⁶ [i]