MinT - Best Known (110−35, 110, s)-Nets in Base 7

Best Known (110−35, 110, s)-Nets in Base 7

(110−35, 110, 206)-Net over F₇ — Constructive and digital

Digital (75, 110, 206)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (19, 36, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 18, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]
2. digital (39, 74, 104)-net over F₇, using
  - trace code for nets [i] based on digital (2, 37, 52)-net over F₄₉, using
    - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(110−35, 110, 1240)-Net over F₇ — Digital

Digital (75, 110, 1240)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−35, 110, 313580)-Net in Base 7 — Upper bound on s

There is no (75, 110, 313581)-net in base 7, because

1 times m-reduction [i] would yield (75, 109, 313581)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 130 527911 265758 904544 370520 409456 727329 253760 093983 961873 717734 115936 005321 192500 214394 626575 > 7¹⁰⁹ [i]