MinT - Best Known (168−69, 168, s)-Nets in Base 3

Best Known (168−69, 168, s)-Nets in Base 3

(168−69, 168, 128)-Net over F₃ — Constructive and digital

Digital (99, 168, 128)-net over F₃, using

4 times m-reduction [i] based on digital (99, 172, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 86, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(168−69, 168, 149)-Net over F₃ — Digital

Digital (99, 168, 149)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−69, 168, 1459)-Net in Base 3 — Upper bound on s

There is no (99, 168, 1460)-net in base 3, because

1 times m-reduction [i] would yield (99, 167, 1460)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 48 391272 029974 832664 148930 473039 244544 680899 595068 320793 298857 757095 927945 792345 > 3¹⁶⁷ [i]