MinT - Best Known (209−43, 209, s)-Nets in Base 3

Best Known (209−43, 209, s)-Nets in Base 3

(209−43, 209, 688)-Net over F₃ — Constructive and digital

Digital (166, 209, 688)-net over F₃, using

3 times m-reduction [i] based on digital (166, 212, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 53, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(209−43, 209, 2087)-Net over F₃ — Digital

Digital (166, 209, 2087)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁹, 2087, F₃, 43) (dual of [2087, 1878, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁹, 2228, F₃, 43) (dual of [2228, 2019, 44]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3²⁰⁷, 2226, F₃, 43) (dual of [2226, 2019, 44]-code), using
    - constructionÂ X applied to C([0,21]) âŠ‚ C([0,18]) [i] based on
      1. linear OA(3¹⁹⁷, 2188, F₃, 43) (dual of [2188, 1991, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
      2. linear OA(3¹⁶⁹, 2188, F₃, 37) (dual of [2188, 2019, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
      3. linear OA(3¹⁰, 38, F₃, 5) (dual of [38, 28, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁰, 39, F₃, 5) (dual of [39, 29, 6]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(209−43, 209, 230779)-Net in Base 3 — Upper bound on s

There is no (166, 209, 230780)-net in base 3, because

1 times m-reduction [i] would yield (166, 208, 230780)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1742 840402 263720 889165 934982 521194 220241 576365 137555 807903 797863 772446 846221 192229 446961 469806 589401 > 3²⁰⁸ [i]