MinT - Best Known (82−25, 82, s)-Nets in Base 7

Best Known (82−25, 82, s)-Nets in Base 7

(82−25, 82, 208)-Net over F₇ — Constructive and digital

Digital (57, 82, 208)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (16, 28, 104)-net over F₇, using
  - trace code for nets [i] based on digital (2, 14, 52)-net over F₄₉, using
    - net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
      - Niederreiter sequence [i]
2. digital (29, 54, 104)-net over F₇, using
  - trace code for nets [i] based on digital (2, 27, 52)-net over F₄₉, using
    - net from sequence [i] based on digital (2, 51)-sequence over F₄₉ (see above)

(82−25, 82, 1273)-Net over F₇ — Digital

Digital (57, 82, 1273)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−25, 82, 446287)-Net in Base 7 — Upper bound on s

There is no (57, 82, 446288)-net in base 7, because

1 times m-reduction [i] would yield (57, 81, 446288)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 283 758013 068950 445034 766261 973998 254287 936076 152920 498700 521041 262209 > 7⁸¹ [i]