MinT - Best Known (208−37, 208, s)-Nets in Base 3

Best Known (208−37, 208, s)-Nets in Base 3

(208−37, 208, 896)-Net over F₃ — Constructive and digital

Digital (171, 208, 896)-net over F₃, using

t-expansion [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]

(208−37, 208, 4581)-Net over F₃ — Digital

Digital (171, 208, 4581)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁸, 4581, F₃, 37) (dual of [4581, 4373, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁸, 6609, F₃, 37) (dual of [6609, 6401, 38]-code), using
  - 4 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]

(208−37, 208, 1158745)-Net in Base 3 — Upper bound on s

There is no (171, 208, 1158746)-net in base 3, because

1 times m-reduction [i] would yield (171, 207, 1158746)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 580 905414 557137 930045 044272 722995 842190 586226 156442 967994 433948 059051 297393 165751 263915 443206 671877 > 3²⁰⁷ [i]