MinT - Best Known (219−42, 219, s)-Nets in Base 3

Best Known (219−42, 219, s)-Nets in Base 3

(219−42, 219, 696)-Net over F₃ — Constructive and digital

Digital (177, 219, 696)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 23, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (154, 196, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 49, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(219−42, 219, 2874)-Net over F₃ — Digital

Digital (177, 219, 2874)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−42, 219, 410330)-Net in Base 3 — Upper bound on s

There is no (177, 219, 410331)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 308 718402 855114 036877 583448 160864 906992 328853 915976 257423 124798 888457 148092 835413 823528 217159 612758 364111 > 3²¹⁹ [i]