MinT - Best Known (54−14, 54, s)-Nets in Base 64

Best Known (54−14, 54, s)-Nets in Base 64

Digital (40, 54, 1198371)-net over F₆₄, using

64¹ times duplication [i] based on digital (39, 53, 1198371)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64⁵³, 1198371, F₆₄, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
  - OA 7-folding and stacking [i] based on linear OA(64⁵³, 8388597, F₆₄, 14) (dual of [8388597, 8388544, 15]-code), using
    - discarding factors / shortening the dual code based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

Digital (40, 54, 7967379)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁵⁴, 7967379, F₆₄, 14) (dual of [7967379, 7967325, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(64⁵⁴, large, F₆₄, 14) (dual of [large, large−54, 15]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(64⁵³, large, F₆₄, 14) (dual of [large, large−53, 15]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

There is no (40, 54, large)-net in base 64, because

12 times m-reduction [i] would yield (40, 42, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]