MinT - Best Known (240−43, 240, s)-Nets in Base 3

Best Known (240−43, 240, s)-Nets in Base 3

(240−43, 240, 1480)-Net over F₃ — Constructive and digital

Digital (197, 240, 1480)-net over F₃, using

t-expansion [i] based on digital (196, 240, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 60, 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]

(240−43, 240, 4838)-Net over F₃ — Digital

Digital (197, 240, 4838)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴⁰, 4838, F₃, 43) (dual of [4838, 4598, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁴⁰, 6609, F₃, 43) (dual of [6609, 6369, 44]-code), using
  - 4 times code embedding in larger space [i] based on linear OA(3²³⁶, 6605, F₃, 43) (dual of [6605, 6369, 44]-code), using
    - constructionÂ X applied to C([0,21]) âŠ‚ C([0,18]) [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(3¹⁹³, 6562, F₃, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (BCH-bound) [i]
      3. linear OA(3¹¹, 43, F₃, 5) (dual of [43, 32, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(240−43, 240, 1168286)-Net in Base 3 — Upper bound on s

There is no (197, 240, 1168287)-net in base 3, because

1 times m-reduction [i] would yield (197, 239, 1168287)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 076431 639118 659999 433618 400026 122015 905062 280674 046569 919909 167382 946722 255910 157421 239230 793320 546273 248229 796375 > 3²³⁹ [i]