MinT - Best Known (140−8, 140, s)-Nets in Base 2

Best Known (140−8, 140, s)-Nets in Base 2

(140−8, 140, 4198398)-Net over F₂ — Constructive and digital

Digital (132, 140, 4198398)-net over F₂, using

t-expansion [i] based on digital (131, 140, 4198398)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2¹⁴⁰, 4198398, F₂, 9, 9) (dual of [(4198398, 9), 37785442, 10]-NRT-code), using
  - appending kth column [i] based on linear OOA(2¹⁴⁰, 4198398, F₂, 8, 9) (dual of [(4198398, 8), 33587044, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2²⁴, 4097, F₂, 8, 4) (dual of [(4097, 8), 32752, 5]-NRT-code), using
        appending 4 arbitrary columns [i] based on linear OOA(2²⁴, 4097, F₂, 4, 4) (dual of [(4097, 4), 16364, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2²⁴, 4097, F₂, 3, 4) (dual of [(4097, 3), 12267, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (20, 24, 4097)-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]

(140−8, 140, large)-Net over F₂ — Digital

Digital (132, 140, large)-net over F₂, using

t-expansion [i] based on digital (130, 140, large)-net over F₂, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁴⁰, large, F₂, 3, 10), using
  - strength reduction [i] based on linear OOA(2¹⁴⁰, large, F₂, 3, 11), using
    - OOAs constructed from BCH codes [i]

(140−8, 140, large)-Net in Base 2 — Upper bound on s

There is no (132, 140, large)-net in base 2, because

6 times m-reduction [i] would yield (132, 134, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]