MinT - Best Known (245−124, 245, s)-Nets in Base 3

Best Known (245−124, 245, s)-Nets in Base 3

(245−124, 245, 78)-Net over F₃ — Constructive and digital

Digital (121, 245, 78)-net over F₃, using

net from sequence [i] based on digital (121, 77)-sequence over F₃, using
- base reduction for sequences [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]

(245−124, 245, 120)-Net over F₃ — Digital

Digital (121, 245, 120)-net over F₃, using

t-expansion [i] based on digital (113, 245, 120)-net over F₃, using
- net from sequence [i] based on digital (113, 119)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(245−124, 245, 859)-Net in Base 3 — Upper bound on s

There is no (121, 245, 860)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 824 764070 687827 313346 347422 435551 344394 070069 426075 325749 604219 259647 453978 297042 155411 685146 480561 365597 120989 273961 > 3²⁴⁵ [i]