MinT - Best Known (189−29, 189, s)-Nets in Base 3

Best Known (189−29, 189, s)-Nets in Base 3

(189−29, 189, 1480)-Net over F₃ — Constructive and digital

Digital (160, 189, 1480)-net over F₃, using

3 times m-reduction [i] based on digital (160, 192, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 48, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(189−29, 189, 11444)-Net over F₃ — Digital

Digital (160, 189, 11444)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸⁹, 11444, F₃, 29) (dual of [11444, 11255, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸⁹, 19745, F₃, 29) (dual of [19745, 19556, 30]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3¹⁸⁷, 19743, F₃, 29) (dual of [19743, 19556, 30]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(21) [i] based on
      1. linear OA(3¹⁷², 19683, F₃, 29) (dual of [19683, 19511, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      2. linear OA(3¹²⁷, 19683, F₃, 22) (dual of [19683, 19556, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      3. linear OA(3¹⁵, 60, F₃, 6) (dual of [60, 45, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁵, 85, F₃, 6) (dual of [85, 70, 7]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(189−29, 189, 7717627)-Net in Base 3 — Upper bound on s

There is no (160, 189, 7717628)-net in base 3, because

1 times m-reduction [i] would yield (160, 188, 7717628)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 499799 709220 109687 643028 451429 652995 053540 024555 960077 894117 245039 633661 336788 279487 159081 > 3¹⁸⁸ [i]