MinT - Best Known (220−35, 220, s)-Nets in Base 3

Best Known (220−35, 220, s)-Nets in Base 3

(220−35, 220, 1480)-Net over F₃ — Constructive and digital

Digital (185, 220, 1480)-net over F₃, using

t-expansion [i] based on digital (184, 220, 1480)-net over F₃, using
- 4 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]

(220−35, 220, 9865)-Net over F₃ — Digital

Digital (185, 220, 9865)-net over F₃, using

3¹ times duplication [i] based on 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]

(220−35, 220, 5027570)-Net in Base 3 — Upper bound on s

There is no (185, 220, 5027571)-net in base 3, because

1 times m-reduction [i] would yield (185, 219, 5027571)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 308 713238 334555 062512 813775 781747 082556 053679 875329 021460 236750 475791 533713 786300 857974 496791 990025 207879 > 3²¹⁹ [i]