MinT - Best Known (165, 165+17, s)-Nets in Base 3

Best Known (165, 165+17, s)-Nets in Base 3

(165, 165+17, 1048597)-Net over F₃ — Constructive and digital

Digital (165, 182, 1048597)-net over F₃, using

3² times duplication [i] based on digital (163, 180, 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 (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]

(165, 165+17, 4119481)-Net over F₃ — Digital

Digital (165, 182, 4119481)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁸², 4119481, F₃, 2, 17) (dual of [(4119481, 2), 8238780, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁸², 4194324, F₃, 2, 17) (dual of [(4194324, 2), 8388466, 18]-NRT-code), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(3¹⁸⁰, 4194323, F₃, 2, 17) (dual of [(4194323, 2), 8388466, 18]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. 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]
      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]

(165, 165+17, large)-Net in Base 3 — Upper bound on s

There is no (165, 182, large)-net in base 3, because

15 times m-reduction [i] would yield (165, 167, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]