MinT - Best Known (162−4, 162, s)-Nets in Base 2

Best Known (162−4, 162, s)-Nets in Base 2

(162−4, 162, large)-Net over F₂ — Constructive and digital

Digital (158, 162, large)-net over F₂, using

t-expansion [i] based on digital (155, 162, large)-net over F₂, using
- 1 times m-reduction [i] based on digital (155, 163, large)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁶³, large, F₂, 8, 8), using
    - appending kth column [i] based on linear OOA(2¹⁶³, large, F₂, 7, 8), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2⁴⁴, 4194305, F₂, 7, 4) (dual of [(4194305, 7), 29360091, 5]-NRT-code), using
        appending 3 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]
        linear OOA(2¹¹⁹, 4194304, F₂, 7, 8) (dual of [(4194304, 7), 29360009, 9]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(2⁴, 3, F₂, 7, 4) (dual of [(3, 7), 17, 5]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(7;17,2) [i]
        linear OOA(2¹¹⁵, 4194301, F₂, 7, 8) (dual of [(4194301, 7), 29359992, 9]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OOA(2¹¹⁵, large, F₂, 2, 8), using
        OOAs constructed from BCH codes [i]

(162−4, 162, large)-Net in Base 2 — Upper bound on s

There is no (158, 162, large)-net in base 2, because

2 times m-reduction [i] would yield (158, 160, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]