MinT - Best Known (169, 169+37, s)-Nets in Base 3

Best Known (169, 169+37, s)-Nets in Base 3

(169, 169+37, 896)-Net over F₃ — Constructive and digital

Digital (169, 206, 896)-net over F₃, using

2 times m-reduction [i] based on digital (169, 208, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 52, 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]

(169, 169+37, 4300)-Net over F₃ — Digital

Digital (169, 206, 4300)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁶, 4300, F₃, 37) (dual of [4300, 4094, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁶, 6607, F₃, 37) (dual of [6607, 6401, 38]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3²⁰⁴, 6605, F₃, 37) (dual of [6605, 6401, 38]-code), using
    - constructionÂ X applied to C([0,18]) âŠ‚ C([0,15]) [i] based on
      1. linear OA(3¹⁹³, 6562, F₃, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
      2. linear OA(3¹⁶¹, 6562, F₃, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
      3. linear OA(3¹¹, 43, F₃, 5) (dual of [43, 32, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(169, 169+37, 1025589)-Net in Base 3 — Upper bound on s

There is no (169, 206, 1025590)-net in base 3, because

1 times m-reduction [i] would yield (169, 205, 1025590)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 64 544655 325628 642177 666961 759082 627608 419096 151184 725642 814896 176336 421613 449719 074518 279523 250221 > 3²⁰⁵ [i]