MinT - Best Known (137, 137+15, s)-Nets in Base 3

Best Known (137, 137+15, s)-Nets in Base 3

Digital (137, 152, 1198371)-net over F₃, using

3² times duplication [i] based on 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]

Digital (137, 152, 2796201)-net over F₃, using

3² times duplication [i] based on digital (135, 150, 2796201)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵⁰, 2796201, F₃, 3, 15) (dual of [(2796201, 3), 8388453, 16]-NRT-code), using
  - OOA 3-folding [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]

There is no (137, 152, large)-net in base 3, because

13 times m-reduction [i] would yield (137, 139, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]