MinT - Best Known (163, 178, s)-Nets in Base 3

Best Known (163, 178, s)-Nets in Base 3

(163, 178, 1198735)-Net over F₃ — Constructive and digital

Digital (163, 178, 1198735)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 28, 364)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²⁸, 364, F₃, 7, 7) (dual of [(364, 7), 2520, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(3²⁸, 1093, F₃, 7) (dual of [1093, 1065, 8]-code), using
      - the BCH-code C(I) with length 1093Â | 3⁷âˆ’1, defining interval IÂ =Â {−3,−1,1,…,9}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
2. digital (135, 150, 1198371)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁵⁰, 1198371, F₃, 15, 15) (dual of [(1198371, 15), 17975415, 16]-NRT-code), using
    - OOA 7-folding and stacking with additional row [i] based on linear OA(3¹⁵⁰, 8388598, F₃, 15) (dual of [8388598, 8388448, 16]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵⁰, large, F₃, 15) (dual of [large, large−150, 16]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(163, 178, large)-Net over F₃ — Digital

Digital (163, 178, large)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁷⁸, large, F₃, 15) (dual of [large, large−178, 16]-code), using
- 28 times code embedding in larger space [i] based on linear OA(3¹⁵⁰, large, F₃, 15) (dual of [large, large−150, 16]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(163, 178, large)-Net in Base 3 — Upper bound on s

There is no (163, 178, large)-net in base 3, because

13 times m-reduction [i] would yield (163, 165, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]