MinT - Best Known (88−27, 88, s)-Nets in Base 7

Best Known (88−27, 88, s)-Nets in Base 7

(88−27, 88, 208)-Net over F₇ — Constructive and digital

Digital (61, 88, 208)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (17, 30, 104)-net over F₇, using
  - trace code for nets [i] based on digital (2, 15, 52)-net over F₄₉, using
    - net from sequence [i] based on digital (2, 51)-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 (see above)

(88−27, 88, 1288)-Net over F₇ — Digital

Digital (61, 88, 1288)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(88−27, 88, 427467)-Net in Base 7 — Upper bound on s

There is no (61, 88, 427468)-net in base 7, because

1 times m-reduction [i] would yield (61, 87, 427468)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 33 384180 458747 625296 480231 510375 958493 964828 052955 864124 363888 030183 589097 > 7⁸⁷ [i]