MinT - Best Known (154−8, 154, s)-Nets in Base 2

Best Known (154−8, 154, s)-Nets in Base 2

(154−8, 154, 4718590)-Net over F₂ — Constructive and digital

Digital (146, 154, 4718590)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁵⁴, 4718590, F₂, 8, 8) (dual of [(4718590, 8), 37748566, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁵⁴, 4718590, F₂, 7, 8) (dual of [(4718590, 7), 33029976, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2³⁹, 524289, F₂, 7, 4) (dual of [(524289, 7), 3669984, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2³⁹, 524289, F₂, 4, 4) (dual of [(524289, 4), 2097117, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2³⁹, 524289, F₂, 3, 4) (dual of [(524289, 3), 1572828, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (35, 39, 524289)-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]

(154−8, 154, large)-Net over F₂ — Digital

Digital (146, 154, large)-net over F₂, using

2² times duplication [i] based on digital (144, 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]

(154−8, 154, large)-Net in Base 2 — Upper bound on s

There is no (146, 154, large)-net in base 2, because

6 times m-reduction [i] would yield (146, 148, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]