MinT - Best Known (64−8, 64, s)-Nets in Base 2

Best Known (64−8, 64, s)-Nets in Base 2

(64−8, 64, 16384)-Net over F₂ — Constructive and digital

Digital (56, 64, 16384)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁶⁴, 16384, F₂, 8, 8) (dual of [(16384, 8), 131008, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2⁶⁴, 65536, F₂, 8) (dual of [65536, 65472, 9]-code), using
  - 1 times truncation [i] based on linear OA(2⁶⁵, 65537, F₂, 9) (dual of [65537, 65472, 10]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 65537Â | 2³²âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(64−8, 64, 21845)-Net over F₂ — Digital

Digital (56, 64, 21845)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁶⁴, 21845, F₂, 8, 8) (dual of [(21845, 8), 174696, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁶⁴, 21845, F₂, 7, 8) (dual of [(21845, 7), 152851, 9]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁶⁴, 21845, F₂, 3, 8) (dual of [(21845, 3), 65471, 9]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(2⁶⁴, 65535, F₂, 8) (dual of [65535, 65471, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(2⁶⁴, 65536, F₂, 8) (dual of [65536, 65472, 9]-code), using
        1 times truncation [i] based on linear OA(2⁶⁵, 65537, F₂, 9) (dual of [65537, 65472, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 65537Â | 2³²âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(64−8, 64, 145049)-Net in Base 2 — Upper bound on s

There is no (56, 64, 145050)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 18 446991 973339 112826 > 2⁶⁴ [i]