MinT - Best Known (67, 67+31, s)-Nets in Base 7

Best Known (67, 67+31, s)-Nets in Base 7

(67, 67+31, 206)-Net over F₇ — Constructive and digital

Digital (67, 98, 206)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (17, 32, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 16, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]
2. digital (35, 66, 104)-net over F₇, using
  - trace code for nets [i] based on digital (2, 33, 52)-net over F₄₉, using
    - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(67, 67+31, 1172)-Net over F₇ — Digital

Digital (67, 98, 1172)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+31, 312298)-Net in Base 7 — Upper bound on s

There is no (67, 98, 312299)-net in base 7, because

1 times m-reduction [i] would yield (67, 97, 312299)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 9430 315209 059651 616570 653922 918863 519641 966026 515714 681545 331321 648018 361202 033351 > 7⁹⁷ [i]