MinT - Best Known (159, 192, s)-Nets in Base 3

Best Known (159, 192, s)-Nets in Base 3

(159, 192, 896)-Net over F₃ — Constructive and digital

Digital (159, 192, 896)-net over F₃, using

t-expansion [i] based on digital (157, 192, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 48, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(159, 192, 5374)-Net over F₃ — Digital

Digital (159, 192, 5374)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹², 5374, F₃, 33) (dual of [5374, 5182, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹², 6616, F₃, 33) (dual of [6616, 6424, 34]-code), using
  - 1 times truncation [i] based on linear OA(3¹⁹³, 6617, F₃, 34) (dual of [6617, 6424, 35]-code), using
    - constructionÂ X applied to C_e(33) âŠ‚ C_e(25) [i] based on
      1. linear OA(3¹⁷⁷, 6561, F₃, 34) (dual of [6561, 6384, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      2. linear OA(3¹³⁷, 6561, F₃, 26) (dual of [6561, 6424, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
      3. linear OA(3¹⁶, 56, F₃, 7) (dual of [56, 40, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(159, 192, 1687095)-Net in Base 3 — Upper bound on s

There is no (159, 192, 1687096)-net in base 3, because

1 times m-reduction [i] would yield (159, 191, 1687096)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13 494627 737477 008848 981135 642759 101311 148435 321783 720469 271119 513802 261664 802347 919331 039233 > 3¹⁹¹ [i]