MinT - Best Known (185, 185+41, s)-Nets in Base 3

Best Known (185, 185+41, s)-Nets in Base 3

(185, 185+41, 896)-Net over F₃ — Constructive and digital

Digital (185, 226, 896)-net over F₃, using

t-expansion [i] based on digital (184, 226, 896)-net over F₃, using
- 2 times m-reduction [i] based on digital (184, 228, 896)-net over F₃, using
  - trace code for nets [i] based on digital (13, 57, 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]

(185, 185+41, 4318)-Net over F₃ — Digital

Digital (185, 226, 4318)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁶, 4318, F₃, 41) (dual of [4318, 4092, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁶, 6579, F₃, 41) (dual of [6579, 6353, 42]-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(3¹, 17, F₃, 1) (dual of [17, 16, 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]

(185, 185+41, 968013)-Net in Base 3 — Upper bound on s

There is no (185, 226, 968014)-net in base 3, because

1 times m-reduction [i] would yield (185, 225, 968014)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 225052 080572 945606 443175 277042 286982 445579 051561 134083 579116 198368 976750 472223 946927 181414 894538 069239 573929 > 3²²⁵ [i]