MinT - Best Known (33, 33+9, s)-Nets in Base 3

Best Known (33, 33+9, s)-Nets in Base 3

Digital (33, 42, 546)-net over F₃, using

net defined by OOA [i] based on linear OOA(3⁴², 546, F₃, 9, 9) (dual of [(546, 9), 4872, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3⁴², 2185, F₃, 9) (dual of [2185, 2143, 10]-code), using
  - discarding factors / shortening the dual code based on linear OA(3⁴², 2186, F₃, 9) (dual of [2186, 2144, 10]-code), using
    - 1 times truncation [i] based on linear OA(3⁴³, 2187, F₃, 10) (dual of [2187, 2144, 11]-code), using
      - an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

Digital (33, 42, 1093)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁴², 1093, F₃, 2, 9) (dual of [(1093, 2), 2144, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3⁴², 2186, F₃, 9) (dual of [2186, 2144, 10]-code), using
  - 1 times truncation [i] based on linear OA(3⁴³, 2187, F₃, 10) (dual of [2187, 2144, 11]-code), using
    - an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

There is no (33, 42, 86000)-net in base 3, because

1 times m-reduction [i] would yield (33, 41, 86000)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36 473147 450661 648001 > 3⁴¹ [i]