MinT - Best Known (172−17, 172, s)-Nets in Base 3

Best Known (172−17, 172, s)-Nets in Base 3

Digital (155, 172, 1048575)-net over F₃, using

3⁶ times duplication [i] based on digital (149, 166, 1048575)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3¹⁶⁶, 1048575, F₃, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
  - OOA 8-folding and stacking with additional row [i] based on linear OA(3¹⁶⁶, 8388601, F₃, 17) (dual of [8388601, 8388435, 18]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹⁶⁶, large, F₃, 17) (dual of [large, large−166, 18]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

Digital (155, 172, 2796203)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷², 2796203, F₃, 3, 17) (dual of [(2796203, 3), 8388437, 18]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(3¹⁶⁶, 2796201, F₃, 3, 17) (dual of [(2796201, 3), 8388437, 18]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3¹⁶⁶, large, F₃, 17) (dual of [large, large−166, 18]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

There is no (155, 172, large)-net in base 3, because

15 times m-reduction [i] would yield (155, 157, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]