MinT - Best Known (164−8, 164, s)-Nets in Base 2

Best Known (164−8, 164, s)-Nets in Base 2

(164−8, 164, large)-Net over F₂ — Constructive and digital

Digital (156, 164, large)-net over F₂, using

t-expansion [i] based on digital (155, 164, large)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2¹⁶⁴, large, F₂, 9, 9), using
  - appending kth column [i] based on linear OOA(2¹⁶⁴, large, F₂, 8, 9), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2⁴⁴, 4194305, F₂, 8, 4) (dual of [(4194305, 8), 33554396, 5]-NRT-code), using
        appending 4 arbitrary columns [i] based on linear OOA(2⁴⁴, 4194305, F₂, 4, 4) (dual of [(4194305, 4), 16777176, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁴⁴, 4194305, F₂, 3, 4) (dual of [(4194305, 3), 12582871, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (40, 44, 4194305)-net over F₂, using
        nets constructed from net-embeddable BCH codes [i]
      2. linear OOA(2¹²⁰, 4194304, F₂, 8, 9) (dual of [(4194304, 8), 33554312, 10]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(2⁴, 3, F₂, 8, 4) (dual of [(3, 8), 20, 5]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(8;20,2) [i]
        linear OOA(2¹¹⁶, 4194301, F₂, 8, 9) (dual of [(4194301, 8), 33554292, 10]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OOA(2¹¹⁶, large, F₂, 2, 9), using
        OOAs constructed from BCH codes [i]

(164−8, 164, large)-Net in Base 2 — Upper bound on s

There is no (156, 164, large)-net in base 2, because

6 times m-reduction [i] would yield (156, 158, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]