MinT - Best Known (229−42, 229, s)-Nets in Base 3

Best Known (229−42, 229, s)-Nets in Base 3

(229−42, 229, 896)-Net over F₃ — Constructive and digital

Digital (187, 229, 896)-net over F₃, using

3 times m-reduction [i] based on digital (187, 232, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 58, 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]

(229−42, 229, 4096)-Net over F₃ — Digital

Digital (187, 229, 4096)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁹, 4096, F₃, 42) (dual of [4096, 3867, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁹, 6582, F₃, 42) (dual of [6582, 6353, 43]-code), using
  - constructionÂ X applied to C([0,21]) âŠ‚ C([0,19]) [i] based on
    1. linear OA(3²²⁵, 6562, F₃, 43) (dual of [6562, 6337, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    2. linear OA(3²⁰⁹, 6562, F₃, 39) (dual of [6562, 6353, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]
    3. linear OA(3⁴, 20, F₃, 2) (dual of [20, 16, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(229−42, 229, 692378)-Net in Base 3 — Upper bound on s

There is no (187, 229, 692379)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 229621 491711 017689 100092 074897 259723 285714 510266 977044 250676 143222 291511 158884 621915 574750 089372 619053 625679 > 3²²⁹ [i]