MinT - Best Known (149−8, 149, s)-Nets in Base 2

Best Known (149−8, 149, s)-Nets in Base 2

(149−8, 149, 4325372)-Net over F₂ — Constructive and digital

Digital (141, 149, 4325372)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹⁴⁹, 4325372, F₂, 8, 8) (dual of [(4325372, 8), 34602827, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹⁴⁹, 4325372, F₂, 7, 8) (dual of [(4325372, 7), 30277455, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2³⁴, 131071, F₂, 7, 4) (dual of [(131071, 7), 917463, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2³⁴, 131071, F₂, 4, 4) (dual of [(131071, 4), 524250, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2³⁴, 131071, F₂, 3, 4) (dual of [(131071, 3), 393179, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (30, 34, 131071)-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]

(149−8, 149, large)-Net over F₂ — Digital

Digital (141, 149, large)-net over F₂, using

3 times m-reduction [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]

(149−8, 149, large)-Net in Base 2 — Upper bound on s

There is no (141, 149, large)-net in base 2, because

6 times m-reduction [i] would yield (141, 143, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]