MinT - Best Known (199, 199+47, s)-Nets in Base 3

Best Known (199, 199+47, s)-Nets in Base 3

(199, 199+47, 896)-Net over F₃ — Constructive and digital

Digital (199, 246, 896)-net over F₃, using

2 times m-reduction [i] based on digital (199, 248, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 62, 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]

(199, 199+47, 3307)-Net over F₃ — Digital

Digital (199, 246, 3307)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴⁶, 3307, F₃, 47) (dual of [3307, 3061, 48]-code), using
- 17 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 16 times 0) [i] based on linear OA(3²⁴⁵, 3289, F₃, 47) (dual of [3289, 3044, 48]-code), using
  - constructionÂ X applied to C_e(46) âŠ‚ C_e(45) [i] based on
    1. linear OA(3²⁴⁵, 3281, F₃, 47) (dual of [3281, 3036, 48]-code), using an extension C_e(46) of the narrow-sense BCH-code C(I) with length 3280Â | 3⁸âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]
    2. linear OA(3²³⁷, 3281, F₃, 46) (dual of [3281, 3044, 47]-code), using an extension C_e(45) of the narrow-sense BCH-code C(I) with length 3280Â | 3⁸âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
    3. linear OA(3⁰, 8, F₃, 0) (dual of [8, 8, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(199, 199+47, 569885)-Net in Base 3 — Upper bound on s

There is no (199, 246, 569886)-net in base 3, because

1 times m-reduction [i] would yield (199, 245, 569886)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 784 717894 249529 998813 738347 013783 744740 254709 994994 582101 730139 924351 541687 303320 762397 724733 496845 454468 905526 774553 > 3²⁴⁵ [i]