MinT - Best Known (168−16, 168, s)-Nets in Base 3

Best Known (168−16, 168, s)-Nets in Base 3

(168−16, 168, 1048597)-Net over F₃ — Constructive and digital

Digital (152, 168, 1048597)-net over F₃, using

3³ times duplication [i] based on digital (149, 165, 1048597)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 14, 22)-net over F₃, using
    - linear code embedding found in a computer search [i]
  2. digital (135, 151, 1048575)-net over F₃, using
    - net defined by OOA [i] based on linear OOA(3¹⁵¹, 1048575, F₃, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
      - OA 8-folding and stacking [i] based on linear OA(3¹⁵¹, 8388600, F₃, 16) (dual of [8388600, 8388449, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁵¹, large, F₃, 16) (dual of [large, large−151, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(168−16, 168, 3506276)-Net over F₃ — Digital

Digital (152, 168, 3506276)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶⁸, 3506276, F₃, 2, 16) (dual of [(3506276, 2), 7012384, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁶⁸, 4194324, F₃, 2, 16) (dual of [(4194324, 2), 8388480, 17]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3¹⁶⁷, 4194324, F₃, 2, 16) (dual of [(4194324, 2), 8388481, 17]-NRT-code), using
    - 1 times NRT-code embedding in larger space [i] based on linear OOA(3¹⁶⁵, 4194323, F₃, 2, 16) (dual of [(4194323, 2), 8388481, 17]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(3¹⁴, 22, F₃, 2, 8) (dual of [(22, 2), 30, 9]-NRT-code), using
        extracting embedded OOA [i] based on digital (6, 14, 22)-net over F₃, using
        linear code embedding found in a computer search [i]
        linear OOA(3¹⁵¹, 4194301, F₃, 2, 16) (dual of [(4194301, 2), 8388451, 17]-NRT-code), using
        OOA 2-folding [i] based on linear OA(3¹⁵¹, 8388602, F₃, 16) (dual of [8388602, 8388451, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁵¹, large, F₃, 16) (dual of [large, large−151, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(168−16, 168, large)-Net in Base 3 — Upper bound on s

There is no (152, 168, large)-net in base 3, because

14 times m-reduction [i] would yield (152, 154, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]