MinT - Best Known (169, 169+45, s)-Nets in Base 3

Best Known (169, 169+45, s)-Nets in Base 3

(169, 169+45, 688)-Net over F₃ — Constructive and digital

Digital (169, 214, 688)-net over F₃, using

2 times m-reduction [i] based on digital (169, 216, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 54, 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]

(169, 169+45, 1909)-Net over F₃ — Digital

Digital (169, 214, 1909)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹⁴, 1909, F₃, 45) (dual of [1909, 1695, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3²¹⁴, 2205, F₃, 45) (dual of [2205, 1991, 46]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3²¹², 2203, F₃, 45) (dual of [2203, 1991, 46]-code), using
    - constructionÂ X applied to C([0,22]) âŠ‚ C([0,21]) [i] based on
      1. linear OA(3²¹¹, 2188, F₃, 45) (dual of [2188, 1977, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,22], and minimum distance dÂ â‰¥ |{−22,−21,…,22}|+1Â =Â 46 (BCH-bound) [i]
      2. 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]
      3. linear OA(3¹, 15, F₃, 1) (dual of [15, 14, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(169, 169+45, 188440)-Net in Base 3 — Upper bound on s

There is no (169, 214, 188441)-net in base 3, because

1 times m-reduction [i] would yield (169, 213, 188441)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 423488 979034 225221 426561 825046 038825 157316 299530 010110 959887 950793 694096 048448 777963 161232 838623 744777 > 3²¹³ [i]