MinT - Best Known (186−40, 186, s)-Nets in Base 3

Best Known (186−40, 186, s)-Nets in Base 3

(186−40, 186, 640)-Net over F₃ — Constructive and digital

Digital (146, 186, 640)-net over F₃, using

2 times m-reduction [i] based on digital (146, 188, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 47, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(186−40, 186, 1545)-Net over F₃ — Digital

Digital (146, 186, 1545)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸⁶, 1545, F₃, 40) (dual of [1545, 1359, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸⁶, 2200, F₃, 40) (dual of [2200, 2014, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(36) [i] based on
    1. linear OA(3¹⁸³, 2187, F₃, 40) (dual of [2187, 2004, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(3¹⁶⁹, 2187, F₃, 37) (dual of [2187, 2018, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(186−40, 186, 113613)-Net in Base 3 — Upper bound on s

There is no (146, 186, 113614)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55539 614270 187268 605434 316479 727323 742862 754087 883330 710464 584719 832582 230096 918479 812009 > 3¹⁸⁶ [i]