MinT - Best Known (190−67, 190, s)-Nets in Base 3

Best Known (190−67, 190, s)-Nets in Base 3

(190−67, 190, 156)-Net over F₃ — Constructive and digital

Digital (123, 190, 156)-net over F₃, using

12 times m-reduction [i] based on digital (123, 202, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 101, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(190−67, 190, 257)-Net over F₃ — Digital

Digital (123, 190, 257)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(190−67, 190, 3523)-Net in Base 3 — Upper bound on s

There is no (123, 190, 3524)-net in base 3, because

1 times m-reduction [i] would yield (123, 189, 3524)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 508549 279450 178194 740013 672289 282368 407163 161487 103519 955167 094174 599896 315016 423163 149705 > 3¹⁸⁹ [i]