MinT - Best Known (184, 219, s)-Nets in Base 3

Best Known (184, 219, s)-Nets in Base 3

(184, 219, 1480)-Net over F₃ — Constructive and digital

Digital (184, 219, 1480)-net over F₃, using

5 times m-reduction [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]

(184, 219, 9865)-Net over F₃ — Digital

Digital (184, 219, 9865)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁹, 9865, F₃, 2, 35) (dual of [(9865, 2), 19511, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3²¹⁹, 19730, F₃, 35) (dual of [19730, 19511, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(28) [i] based on
    1. linear OA(3²⁰⁸, 19683, F₃, 35) (dual of [19683, 19475, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    2. linear OA(3¹⁷², 19683, F₃, 29) (dual of [19683, 19511, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(3¹¹, 47, F₃, 5) (dual of [47, 36, 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]

(184, 219, 4712942)-Net in Base 3 — Upper bound on s

There is no (184, 219, 4712943)-net in base 3, because

1 times m-reduction [i] would yield (184, 218, 4712943)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 102 904517 979078 117364 937756 179088 177598 896010 754190 472501 410581 581157 100628 246766 520005 066879 303646 151615 > 3²¹⁸ [i]