MinT - Best Known (213−68, 213, s)-Nets in Base 3

Best Known (213−68, 213, s)-Nets in Base 3

(213−68, 213, 167)-Net over F₃ — Constructive and digital

Digital (145, 213, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 43, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]
2. digital (102, 170, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 85, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(213−68, 213, 385)-Net over F₃ — Digital

Digital (145, 213, 385)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−68, 213, 6565)-Net in Base 3 — Upper bound on s

There is no (145, 213, 6566)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 424689 838510 520432 813982 351904 656336 990929 616499 709143 426273 745287 037058 422530 991469 334273 951254 351213 > 3²¹³ [i]