MinT - Best Known (187−17, 187, s)-Nets in Base 3

Best Known (187−17, 187, s)-Nets in Base 3

(187−17, 187, 1048619)-Net over F₃ — Constructive and digital

Digital (170, 187, 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 (149, 166, 1048575)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶⁶, 1048575, F₃, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(3¹⁶⁶, 8388601, F₃, 17) (dual of [8388601, 8388435, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶⁶, large, F₃, 17) (dual of [large, large−166, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(187−17, 187, 4194355)-Net over F₃ — Digital

Digital (170, 187, 4194355)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁸⁷, 4194355, F₃, 2, 17) (dual of [(4194355, 2), 8388523, 18]-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(3¹⁶⁶, 4194301, F₃, 2, 17) (dual of [(4194301, 2), 8388436, 18]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁶⁶, 8388602, F₃, 17) (dual of [8388602, 8388436, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶⁶, large, F₃, 17) (dual of [large, large−166, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(187−17, 187, large)-Net in Base 3 — Upper bound on s

There is no (170, 187, large)-net in base 3, because

15 times m-reduction [i] would yield (170, 172, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]