MinT - Best Known (225−39, 225, s)-Nets in Base 3

Best Known (225−39, 225, s)-Nets in Base 3

(225−39, 225, 1480)-Net over F₃ — Constructive and digital

Digital (186, 225, 1480)-net over F₃, using

3¹ times duplication [i] based on digital (185, 224, 1480)-net over F₃, using
- t-expansion [i] based on digital (184, 224, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 56, 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]

(225−39, 225, 5632)-Net over F₃ — Digital

Digital (186, 225, 5632)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁵, 5632, F₃, 39) (dual of [5632, 5407, 40]-code), using
- discarding factors / shortening the dual code 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]

(225−39, 225, 1671818)-Net in Base 3 — Upper bound on s

There is no (186, 225, 1671819)-net in base 3, because

1 times m-reduction [i] would yield (186, 224, 1671819)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 75017 481823 192557 989420 786414 389178 509599 020488 299421 910376 752474 416187 836183 189737 917450 141548 431277 751595 > 3²²⁴ [i]