MinT - Best Known (86−27, 86, s)-Nets in Base 7

Best Known (86−27, 86, s)-Nets in Base 7

(86−27, 86, 206)-Net over F₇ — Constructive and digital

Digital (59, 86, 206)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (15, 28, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 14, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]
2. digital (31, 58, 104)-net over F₇, using
  - trace code for nets [i] based on digital (2, 29, 52)-net over F₄₉, using
    - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(86−27, 86, 1111)-Net over F₇ — Digital

Digital (59, 86, 1111)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−27, 86, 316872)-Net in Base 7 — Upper bound on s

There is no (59, 86, 316873)-net in base 7, because

1 times m-reduction [i] would yield (59, 85, 316873)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 681301 752205 469362 827537 252209 142357 062334 604171 585625 116359 254281 978903 > 7⁸⁵ [i]