MinT - Best Known (164, 164+79, s)-Nets in Base 3

Best Known (164, 164+79, s)-Nets in Base 3

(164, 164+79, 168)-Net over F₃ — Constructive and digital

Digital (164, 243, 168)-net over F₃, using

3¹ times duplication [i] based on digital (163, 242, 168)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 50, 20)-net over F₃, using
    - net from sequence [i] based on digital (11, 19)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 9, N(F)Â =Â 19, and 1 place with degree 3 [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 (113, 192, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 96, 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]

(164, 164+79, 407)-Net over F₃ — Digital

Digital (164, 243, 407)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(164, 164+79, 6992)-Net in Base 3 — Upper bound on s

There is no (164, 243, 6993)-net in base 3, because

1 times m-reduction [i] would yield (164, 242, 6993)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 190705 814400 143041 841631 511657 953146 117890 270999 274281 959429 208626 477756 146382 476472 217490 282540 049836 738680 830283 > 3²⁴² [i]