MinT - Best Known (52−6, 52, s)-Nets in Base 3

Best Known (52−6, 52, s)-Nets in Base 3

(52−6, 52, 531441)-Net over F₃ — Constructive and digital

Digital (46, 52, 531441)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁵², 531441, F₃, 6, 6) (dual of [(531441, 6), 3188594, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁵², 531441, F₃, 5, 6) (dual of [(531441, 5), 2657153, 7]-NRT-code), using
  - OA 3-folding and stacking [i] based on linear OA(3⁵², 1594323, F₃, 6) (dual of [1594323, 1594271, 7]-code), using
    - 1 times truncation [i] based on linear OA(3⁵³, 1594324, F₃, 7) (dual of [1594324, 1594271, 8]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 1594324Â | 3²⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(52−6, 52, 1340657)-Net over F₃ — Digital

Digital (46, 52, 1340657)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁵², 1340657, F₃, 6, 6) (dual of [(1340657, 6), 8043890, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁵², 1340657, F₃, 5, 6) (dual of [(1340657, 5), 6703233, 7]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵², 1340657, F₃, 6) (dual of [1340657, 1340605, 7]-code), using
    - discarding factors / shortening the dual code based on linear OA(3⁵², 1594323, F₃, 6) (dual of [1594323, 1594271, 7]-code), using
      - 1 times truncation [i] based on linear OA(3⁵³, 1594324, F₃, 7) (dual of [1594324, 1594271, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 1594324Â | 3²⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(52−6, 52, large)-Net in Base 3 — Upper bound on s

There is no (46, 52, large)-net in base 3, because

4 times m-reduction [i] would yield (46, 48, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]