MinT - Best Known (137, 137+8, s)-Nets in Base 2

Best Known (137, 137+8, s)-Nets in Base 2

(137, 137+8, 4227068)-Net over F₂ — Constructive and digital

Digital (137, 145, 4227068)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁴⁵, 4227068, F₂, 8, 8) (dual of [(4227068, 8), 33816399, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁴⁵, 4227068, F₂, 7, 8) (dual of [(4227068, 7), 29589331, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2³⁰, 32767, F₂, 7, 4) (dual of [(32767, 7), 229339, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2³⁰, 32767, F₂, 4, 4) (dual of [(32767, 4), 131038, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2³⁰, 32767, F₂, 3, 4) (dual of [(32767, 3), 98271, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (26, 30, 32767)-net over F₂, using
        nets constructed from net-embeddable BCH codes [i]
    2. 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]

(137, 137+8, large)-Net over F₂ — Digital

Digital (137, 145, large)-net over F₂, using

2⁵ times duplication [i] based on 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]

(137, 137+8, large)-Net in Base 2 — Upper bound on s

There is no (137, 145, large)-net in base 2, because

6 times m-reduction [i] would yield (137, 139, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]