MinT - Best Known (192−39, 192, s)-Nets in Base 3

Best Known (192−39, 192, s)-Nets in Base 3

(192−39, 192, 688)-Net over F₃ — Constructive and digital

Digital (153, 192, 688)-net over F₃, using

t-expansion [i] based on digital (151, 192, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 48, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(192−39, 192, 2093)-Net over F₃ — Digital

Digital (153, 192, 2093)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹², 2093, F₃, 39) (dual of [2093, 1901, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹², 2224, F₃, 39) (dual of [2224, 2032, 40]-code), using
  - 1 times truncation [i] based on linear OA(3¹⁹³, 2225, F₃, 40) (dual of [2225, 2032, 41]-code), using
    - constructionÂ X applied to C_e(39) âŠ‚ C_e(33) [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(3¹⁵⁵, 2187, F₃, 34) (dual of [2187, 2032, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      3. linear OA(3¹⁰, 38, F₃, 5) (dual of [38, 28, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁰, 39, F₃, 5) (dual of [39, 29, 6]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(192−39, 192, 248014)-Net in Base 3 — Upper bound on s

There is no (153, 192, 248015)-net in base 3, because

1 times m-reduction [i] would yield (153, 191, 248015)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13 494965 182816 071679 778983 143653 389681 048942 573830 073033 753658 841883 636253 998869 164024 167451 > 3¹⁹¹ [i]