MinT - Best Known (196−22, 196, s)-Nets in Base 3

Best Known (196−22, 196, s)-Nets in Base 3

(196−22, 196, 144946)-Net over F₃ — Constructive and digital

Digital (174, 196, 144946)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (161, 183, 144938)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁸³, 144938, F₃, 22, 22) (dual of [(144938, 22), 3188453, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(3¹⁸³, 1594318, F₃, 22) (dual of [1594318, 1594135, 23]-code), using
      - discarding factors / shortening the dual code 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]

(196−22, 196, 493043)-Net over F₃ — Digital

Digital (174, 196, 493043)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹⁶, 493043, F₃, 3, 22) (dual of [(493043, 3), 1478933, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁹⁶, 531462, F₃, 3, 22) (dual of [(531462, 3), 1594190, 23]-NRT-code), using
  - 3² times duplication [i] based on linear OOA(3¹⁹⁴, 531462, F₃, 3, 22) (dual of [(531462, 3), 1594192, 23]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(3¹⁹⁴, 1594386, F₃, 22) (dual of [1594386, 1594192, 23]-code), using
      - constructionÂ X applied to C_e(21) âŠ‚ C_e(15) [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]
        linear OA(3¹³¹, 1594323, F₃, 16) (dual of [1594323, 1594192, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 1594322Â = 3¹³âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [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]

(196−22, 196, large)-Net in Base 3 — Upper bound on s

There is no (174, 196, large)-net in base 3, because

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