MinT - Best Known (246−124, 246, s)-Nets in Base 3

Best Known (246−124, 246, s)-Nets in Base 3

(246−124, 246, 78)-Net over F₃ — Constructive and digital

Digital (122, 246, 78)-net over F₃, using

t-expansion [i] based on digital (121, 246, 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]

(246−124, 246, 120)-Net over F₃ — Digital

Digital (122, 246, 120)-net over F₃, using

t-expansion [i] based on digital (113, 246, 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]

(246−124, 246, 875)-Net in Base 3 — Upper bound on s

There is no (122, 246, 876)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2406 852605 589147 881303 417608 656865 026766 835950 795821 569605 998222 393660 396799 483219 455475 825371 672771 615394 390118 712009 > 3²⁴⁶ [i]