MinT - Best Known (156−9, 156, s)-Nets in Base 2

Best Known (156−9, 156, s)-Nets in Base 2

(156−9, 156, 5242878)-Net over F₂ — Constructive and digital

Digital (147, 156, 5242878)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁵⁶, 5242878, F₂, 9, 9) (dual of [(5242878, 9), 47185746, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁵⁶, 5242878, F₂, 8, 9) (dual of [(5242878, 8), 41942868, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁴⁰, 1048577, F₂, 8, 4) (dual of [(1048577, 8), 8388576, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2⁴⁰, 1048577, F₂, 4, 4) (dual of [(1048577, 4), 4194268, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁴⁰, 1048577, F₂, 3, 4) (dual of [(1048577, 3), 3145691, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (36, 40, 1048577)-net over F₂, using
        nets constructed from net-embeddable BCH codes [i]
    2. 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]

(156−9, 156, large)-Net over F₂ — Digital

Digital (147, 156, large)-net over F₂, using

2⁴ times duplication [i] based on digital (143, 152, large)-net over F₂, using
- t-expansion [i] based on digital (141, 152, large)-net over F₂, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁵², large, F₂, 3, 11), using
    - 4 times NRT-code embedding in larger space [i] based on linear OOA(2¹⁴⁰, large, F₂, 3, 11), using
      - OOAs constructed from BCH codes [i]

(156−9, 156, large)-Net in Base 2 — Upper bound on s

There is no (147, 156, large)-net in base 2, because

7 times m-reduction [i] would yield (147, 149, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]