MinT - Best Known (123, 123+75, s)-Nets in Base 3

Best Known (123, 123+75, s)-Nets in Base 3

(123, 123+75, 156)-Net over F₃ — Constructive and digital

Digital (123, 198, 156)-net over F₃, using

4 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]

(123, 123+75, 217)-Net over F₃ — Digital

Digital (123, 198, 217)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+75, 2506)-Net in Base 3 — Upper bound on s

There is no (123, 198, 2507)-net in base 3, because

1 times m-reduction [i] would yield (123, 197, 2507)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9976 454239 305812 390783 481088 902372 537145 776980 156485 812922 698438 359710 374351 697157 961238 917263 > 3¹⁹⁷ [i]