MinT - Best Known (172−16, 172, s)-Nets in Base 3

Best Known (172−16, 172, s)-Nets in Base 3

(172−16, 172, 1048619)-Net over F₃ — Constructive and digital

Digital (156, 172, 1048619)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 21, 44)-net over F₃, using
  - (u,Â u+v)-construction [i] based on
    1. digital (3, 7, 26)-net over F₃, using
      - nets constructed from net-embeddable BCH codes [i]
    2. 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]

(172−16, 172, 4194355)-Net over F₃ — Digital

Digital (156, 172, 4194355)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷², 4194355, F₃, 2, 16) (dual of [(4194355, 2), 8388538, 17]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3²¹, 54, F₃, 2, 8) (dual of [(54, 2), 87, 9]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²¹, 54, F₃, 8) (dual of [54, 33, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹, 80, F₃, 8) (dual of [80, 59, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
  2. 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]

(172−16, 172, large)-Net in Base 3 — Upper bound on s

There is no (156, 172, large)-net in base 3, because

14 times m-reduction [i] would yield (156, 158, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]