MinT - Best Known (219−71, 219, s)-Nets in Base 3

Best Known (219−71, 219, s)-Nets in Base 3

(219−71, 219, 164)-Net over F₃ — Constructive and digital

Digital (148, 219, 164)-net over F₃, using

3¹ times duplication [i] based on digital (147, 218, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 42, 16)-net over F₃, using
    - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
        an explicitly constructive algebraic function field [i]
  2. digital (105, 176, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 88, 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]

(219−71, 219, 375)-Net over F₃ — Digital

Digital (148, 219, 375)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−71, 219, 6482)-Net in Base 3 — Upper bound on s

There is no (148, 219, 6483)-net in base 3, because

1 times m-reduction [i] would yield (148, 218, 6483)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 103 389441 887108 080104 556006 481696 371410 905579 295225 985439 339051 692570 379606 329460 187640 400383 802988 921771 > 3²¹⁸ [i]