MinT - Best Known (180−27, 180, s)-Nets in Base 3

Best Known (180−27, 180, s)-Nets in Base 3

Digital (153, 180, 4542)-net over F₃, using

net defined by OOA [i] based on linear OOA(3¹⁸⁰, 4542, F₃, 27, 27) (dual of [(4542, 27), 122454, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3¹⁸⁰, 59047, F₃, 27) (dual of [59047, 58867, 28]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹⁸⁰, 59048, F₃, 27) (dual of [59048, 58868, 28]-code), using
    - 1 times truncation [i] based on linear OA(3¹⁸¹, 59049, F₃, 28) (dual of [59049, 58868, 29]-code), using
      - an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

Digital (153, 180, 19682)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁸⁰, 19682, F₃, 3, 27) (dual of [(19682, 3), 58866, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3¹⁸⁰, 59046, F₃, 27) (dual of [59046, 58866, 28]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹⁸⁰, 59048, F₃, 27) (dual of [59048, 58868, 28]-code), using
    - 1 times truncation [i] based on linear OA(3¹⁸¹, 59049, F₃, 28) (dual of [59049, 58868, 29]-code), using
      - an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 59048Â = 3¹⁰âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

There is no (153, 180, large)-net in base 3, because

25 times m-reduction [i] would yield (153, 155, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]