MinT - Best Known (66−21, 66, s)-Nets in Base 7

Best Known (66−21, 66, s)-Nets in Base 7

(66−21, 66, 204)-Net over F₇ — Constructive and digital

Digital (45, 66, 204)-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 (23, 44, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 22, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉ (see above)

(66−21, 66, 861)-Net over F₇ — Digital

Digital (45, 66, 861)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(66−21, 66, 234936)-Net in Base 7 — Upper bound on s

There is no (45, 66, 234937)-net in base 7, because

1 times m-reduction [i] would yield (45, 65, 234937)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 8 538422 691967 432427 628454 130075 231361 492383 727929 592673 > 7⁶⁵ [i]