MinT - Best Known (56−6, 56, s)-Nets in Base 3

Best Known (56−6, 56, s)-Nets in Base 3

(56−6, 56, 1594323)-Net over F₃ — Constructive and digital

Digital (50, 56, 1594323)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁵⁶, 1594323, F₃, 6, 6) (dual of [(1594323, 6), 9565882, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁵⁶, 1594323, F₃, 5, 6) (dual of [(1594323, 5), 7971559, 7]-NRT-code), using
  - OA 3-folding and stacking [i] based on linear OA(3⁵⁶, 4782969, F₃, 6) (dual of [4782969, 4782913, 7]-code), using
    - 1 times truncation [i] based on linear OA(3⁵⁷, 4782970, F₃, 7) (dual of [4782970, 4782913, 8]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(56−6, 56, 4021978)-Net over F₃ — Digital

Digital (50, 56, 4021978)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁵⁶, 4021978, F₃, 6, 6) (dual of [(4021978, 6), 24131812, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁵⁶, 4021978, F₃, 5, 6) (dual of [(4021978, 5), 20109834, 7]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵⁶, 4021978, F₃, 6) (dual of [4021978, 4021922, 7]-code), using
    - discarding factors / shortening the dual code based on linear OA(3⁵⁶, 4782969, F₃, 6) (dual of [4782969, 4782913, 7]-code), using
      - 1 times truncation [i] based on linear OA(3⁵⁷, 4782970, F₃, 7) (dual of [4782970, 4782913, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(56−6, 56, large)-Net in Base 3 — Upper bound on s

There is no (50, 56, large)-net in base 3, because

4 times m-reduction [i] would yield (50, 52, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]