MinT - Best Known (195−21, 195, s)-Nets in Base 3

Best Known (195−21, 195, s)-Nets in Base 3

(195−21, 195, 159442)-Net over F₃ — Constructive and digital

Digital (174, 195, 159442)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 13, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (161, 182, 159432)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁸², 159432, F₃, 21, 21) (dual of [(159432, 21), 3347890, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(3¹⁸², 1594321, F₃, 21) (dual of [1594321, 1594139, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁸², 1594322, F₃, 21) (dual of [1594322, 1594140, 22]-code), using
        1 times truncation [i] based on linear OA(3¹⁸³, 1594323, F₃, 22) (dual of [1594323, 1594140, 23]-code), using
        an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 1594322Â = 3¹³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(195−21, 195, 531462)-Net over F₃ — Digital

Digital (174, 195, 531462)-net over F₃, using

3¹ times duplication [i] based on digital (173, 194, 531462)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹⁴, 531462, F₃, 3, 21) (dual of [(531462, 3), 1594192, 22]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3¹⁹⁴, 1594386, F₃, 21) (dual of [1594386, 1594192, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹⁹⁴, 1594387, F₃, 21) (dual of [1594387, 1594193, 22]-code), using
      - constructionÂ X applied to C([0,10]) âŠ‚ C([0,7]) [i] based on
        linear OA(3¹⁸³, 1594324, F₃, 21) (dual of [1594324, 1594141, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324Â | 3²⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(3¹³¹, 1594324, F₃, 15) (dual of [1594324, 1594193, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324Â | 3²⁶âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
        linear OA(3¹¹, 63, F₃, 5) (dual of [63, 52, 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]

(195−21, 195, large)-Net in Base 3 — Upper bound on s

There is no (174, 195, large)-net in base 3, because

19 times m-reduction [i] would yield (174, 176, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]