MinT - Best Known (185, 224, s)-Nets in Base 3

Best Known (185, 224, s)-Nets in Base 3

(185, 224, 1480)-Net over F₃ — Constructive and digital

Digital (185, 224, 1480)-net over F₃, using

t-expansion [i] based on digital (184, 224, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 56, 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]

(185, 224, 5466)-Net over F₃ — Digital

Digital (185, 224, 5466)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁴, 5466, F₃, 39) (dual of [5466, 5242, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁴, 6616, F₃, 39) (dual of [6616, 6392, 40]-code), using
  - 1 times truncation [i] based on linear OA(3²²⁵, 6617, F₃, 40) (dual of [6617, 6392, 41]-code), using
    - constructionÂ X applied to C_e(39) âŠ‚ C_e(31) [i] based on
      1. linear OA(3²⁰⁹, 6561, F₃, 40) (dual of [6561, 6352, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
      2. linear OA(3¹⁶⁹, 6561, F₃, 32) (dual of [6561, 6392, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [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]

(185, 224, 1577891)-Net in Base 3 — Upper bound on s

There is no (185, 224, 1577892)-net in base 3, because

1 times m-reduction [i] would yield (185, 223, 1577892)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25005 771166 071980 106160 261777 452514 629608 655870 216959 144604 148047 961665 089205 476623 253603 846373 621273 637425 > 3²²³ [i]