MinT - Best Known (38, 38+6, s)-Nets in Base 3

Best Known (38, 38+6, s)-Nets in Base 3

(38, 38+6, 59049)-Net over F₃ — Constructive and digital

Digital (38, 44, 59049)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁴⁴, 59049, F₃, 6, 6) (dual of [(59049, 6), 354250, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁴⁴, 59049, F₃, 5, 6) (dual of [(59049, 5), 295201, 7]-NRT-code), using
  - OA 3-folding and stacking [i] based on linear OA(3⁴⁴, 177147, F₃, 6) (dual of [177147, 177103, 7]-code), using
    - 1 times truncation [i] based on linear OA(3⁴⁵, 177148, F₃, 7) (dual of [177148, 177103, 8]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(38, 38+6, 148959)-Net over F₃ — Digital

Digital (38, 44, 148959)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁴⁴, 148959, F₃, 6, 6) (dual of [(148959, 6), 893710, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(3⁴⁴, 148959, F₃, 5, 6) (dual of [(148959, 5), 744751, 7]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁴⁴, 148959, F₃, 6) (dual of [148959, 148915, 7]-code), using
    - discarding factors / shortening the dual code based on linear OA(3⁴⁴, 177147, F₃, 6) (dual of [177147, 177103, 7]-code), using
      - 1 times truncation [i] based on linear OA(3⁴⁵, 177148, F₃, 7) (dual of [177148, 177103, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 177148Â | 3²²âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(38, 38+6, large)-Net in Base 3 — Upper bound on s

There is no (38, 44, large)-net in base 3, because

4 times m-reduction [i] would yield (38, 40, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]