MinT - Best Known (139−9, 139, s)-Nets in Base 2

Best Known (139−9, 139, s)-Nets in Base 2

(139−9, 139, 4196350)-Net over F₂ — Constructive and digital

Digital (130, 139, 4196350)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹³⁹, 4196350, F₂, 9, 9) (dual of [(4196350, 9), 37767011, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹³⁹, 4196350, F₂, 8, 9) (dual of [(4196350, 8), 33570661, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2²³, 2049, F₂, 8, 4) (dual of [(2049, 8), 16369, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2²³, 2049, F₂, 4, 4) (dual of [(2049, 4), 8173, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2²³, 2049, F₂, 3, 4) (dual of [(2049, 3), 6124, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (19, 23, 2049)-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]

(139−9, 139, large)-Net over F₂ — Digital

Digital (130, 139, large)-net over F₂, using

1 times m-reduction [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]

(139−9, 139, large)-Net in Base 2 — Upper bound on s

There is no (130, 139, large)-net in base 2, because

7 times m-reduction [i] would yield (130, 132, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]