MinT - Best Known (75−27, 75, s)-Nets in Base 7

Best Known (75−27, 75, s)-Nets in Base 7

(75−27, 75, 122)-Net over F₇ — Constructive and digital

Digital (48, 75, 122)-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 (29, 56, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 28, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(75−27, 75, 495)-Net over F₇ — Digital

Digital (48, 75, 495)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−27, 75, 61059)-Net in Base 7 — Upper bound on s

There is no (48, 75, 61060)-net in base 7, because

1 times m-reduction [i] would yield (48, 74, 61060)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 344 571515 942890 121714 862977 660584 315603 441212 107638 990861 724473 > 7⁷⁴ [i]