MinT - Best Known (27−3, 27, s)-Nets in Base 64

Best Known (27−3, 27, s)-Nets in Base 64

(27−3, 27, large)-Net over F₆₄ — Constructive and digital

Digital (24, 27, large)-net over F₆₄, using

64³ times duplication [i] based on digital (21, 24, large)-net over F₆₄, using
- t-expansion [i] based on digital (19, 24, large)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (3, 5, large)-net over F₆₄, using
      - kÂ =Â 2 construction [i]
    2. digital (14, 19, 4194366)-net over F₆₄, using
      - (u,Â u+v)-construction [i] based on
        digital (0, 2, 65)-net over F₆₄, using
        kÂ =Â 2 construction [i]
        digital (12, 17, 4194301)-net over F₆₄, using
        net defined by OOA [i] based on linear OOA(64¹⁷, 4194301, F₆₄, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(64¹⁷, large, F₆₄, 5) (dual of [large, large−17, 6]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]

(27−3, 27, large)-Net in Base 64 — Upper bound on s

There is no (24, 27, large)-net in base 64, because

1 times m-reduction [i] would yield (24, 26, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]