MinT - Best Known (227−39, 227, s)-Nets in Base 3

Best Known (227−39, 227, s)-Nets in Base 3

(227−39, 227, 1480)-Net over F₃ — Constructive and digital

Digital (188, 227, 1480)-net over F₃, using

t-expansion [i] based on digital (187, 227, 1480)-net over F₃, using
- 1 times m-reduction [i] based on digital (187, 228, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 57, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(227−39, 227, 5979)-Net over F₃ — Digital

Digital (188, 227, 5979)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁷, 5979, F₃, 39) (dual of [5979, 5752, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²⁷, 6628, F₃, 39) (dual of [6628, 6401, 40]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3²²⁵, 6626, F₃, 39) (dual of [6626, 6401, 40]-code), using
    - constructionÂ X applied to C([0,19]) âŠ‚ C([0,15]) [i] based on
      1. 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]
      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(3¹⁶, 64, F₃, 7) (dual of [64, 48, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(227−39, 227, 1876778)-Net in Base 3 — Upper bound on s

There is no (188, 227, 1876779)-net in base 3, because

1 times m-reduction [i] would yield (188, 226, 1876779)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 675159 223676 727877 839212 535078 656797 143552 632218 369713 253421 691042 318100 716973 390188 281942 046255 571648 055211 > 3²²⁶ [i]