MinT - Best Known (68−21, 68, s)-Nets in Base 7

Best Known (68−21, 68, s)-Nets in Base 7

(68−21, 68, 206)-Net over F₇ — Constructive and digital

Digital (47, 68, 206)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (12, 22, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 11, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]
2. digital (25, 46, 104)-net over F₇, using
  - trace code for nets [i] based on digital (2, 23, 52)-net over F₄₉, using
    - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(68−21, 68, 1044)-Net over F₇ — Digital

Digital (47, 68, 1044)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−21, 68, 346716)-Net in Base 7 — Upper bound on s

There is no (47, 68, 346717)-net in base 7, because

1 times m-reduction [i] would yield (47, 67, 346717)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 418 386897 211038 908580 747566 206803 489194 728867 405064 052169 > 7⁶⁷ [i]