MinT - Best Known (81−12, 81, s)-Nets in Base 4

Best Known (81−12, 81, s)-Nets in Base 4

Digital (69, 81, 43690)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁸¹, 43690, F₄, 12, 12) (dual of [(43690, 12), 524199, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(4⁸¹, 262140, F₄, 12) (dual of [262140, 262059, 13]-code), using
  - discarding factors / shortening the dual code based on linear OA(4⁸¹, 262143, F₄, 12) (dual of [262143, 262062, 13]-code), using
    - 1 times truncation [i] based on linear OA(4⁸², 262144, F₄, 13) (dual of [262144, 262062, 14]-code), using
      - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

Digital (69, 81, 131071)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4⁸¹, 131071, F₄, 2, 12) (dual of [(131071, 2), 262061, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4⁸¹, 262142, F₄, 12) (dual of [262142, 262061, 13]-code), using
  - discarding factors / shortening the dual code based on linear OA(4⁸¹, 262143, F₄, 12) (dual of [262143, 262062, 13]-code), using
    - 1 times truncation [i] based on linear OA(4⁸², 262144, F₄, 13) (dual of [262144, 262062, 14]-code), using
      - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

There is no (69, 81, large)-net in base 4, because

10 times m-reduction [i] would yield (69, 71, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]