MinT - Best Known (219−21, 219, s)-Nets in Base 3

Best Known (219−21, 219, s)-Nets in Base 3

Digital (198, 219, 838860)-net over F₃, using

3⁹ times duplication [i] based on digital (189, 210, 838860)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3²¹⁰, 838860, F₃, 21, 21) (dual of [(838860, 21), 17615850, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(3²¹⁰, 8388601, F₃, 21) (dual of [8388601, 8388391, 22]-code), using
    - discarding factors / shortening the dual code based on linear OA(3²¹⁰, large, F₃, 21) (dual of [large, large−210, 22]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

Digital (198, 219, 2796204)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁹, 2796204, F₃, 3, 21) (dual of [(2796204, 3), 8388393, 22]-NRT-code), using
- 3 times NRT-code embedding in larger space [i] based on linear OOA(3²¹⁰, 2796201, F₃, 3, 21) (dual of [(2796201, 3), 8388393, 22]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3²¹⁰, large, F₃, 21) (dual of [large, large−210, 22]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

There is no (198, 219, large)-net in base 3, because

19 times m-reduction [i] would yield (198, 200, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]