MinT - Best Known (200, 243, s)-Nets in Base 3

Best Known (200, 243, s)-Nets in Base 3

(200, 243, 1480)-Net over F₃ — Constructive and digital

Digital (200, 243, 1480)-net over F₃, using

t-expansion [i] based on digital (199, 243, 1480)-net over F₃, using
- 1 times m-reduction [i] based on digital (199, 244, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 61, 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]

(200, 243, 5246)-Net over F₃ — Digital

Digital (200, 243, 5246)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴³, 5246, F₃, 43) (dual of [5246, 5003, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁴³, 6619, F₃, 43) (dual of [6619, 6376, 44]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3²⁴¹, 6617, F₃, 43) (dual of [6617, 6376, 44]-code), using
    - constructionÂ X applied to C_e(42) âŠ‚ C_e(34) [i] based on
      1. linear OA(3²²⁵, 6561, F₃, 43) (dual of [6561, 6336, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
      2. linear OA(3¹⁸⁵, 6561, F₃, 35) (dual of [6561, 6376, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
      3. linear OA(3¹⁶, 56, F₃, 7) (dual of [56, 40, 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]

(200, 243, 1366817)-Net in Base 3 — Upper bound on s

There is no (200, 243, 1366818)-net in base 3, because

1 times m-reduction [i] would yield (200, 242, 1366818)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 063456 452452 922718 173606 483846 341619 145424 377862 667238 699467 973920 556438 517673 984071 297472 354700 828162 771302 798781 > 3²⁴² [i]