MinT - Best Known (163, 163+81, s)-Nets in Base 3

Best Known (163, 163+81, s)-Nets in Base 3

(163, 163+81, 164)-Net over F₃ — Constructive and digital

Digital (163, 244, 164)-net over F₃, using

3¹ times duplication [i] based on digital (162, 243, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 47, 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 (115, 196, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 98, 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]

(163, 163+81, 383)-Net over F₃ — Digital

Digital (163, 244, 383)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(163, 163+81, 6201)-Net in Base 3 — Upper bound on s

There is no (163, 244, 6202)-net in base 3, because

1 times m-reduction [i] would yield (163, 243, 6202)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 87 261810 306675 370342 722040 104192 628295 113104 114921 248085 092359 064112 992438 514317 251495 706934 588085 464886 407324 027153 > 3²⁴³ [i]