MinT - Best Known (135−8, 135, s)-Nets in Base 2

Best Known (135−8, 135, s)-Nets in Base 2

(135−8, 135, 4195326)-Net over F₂ — Constructive and digital

Digital (127, 135, 4195326)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹³⁵, 4195326, F₂, 8, 8) (dual of [(4195326, 8), 33562473, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹³⁵, 4195326, F₂, 7, 8) (dual of [(4195326, 7), 29367147, 9]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2²⁰, 1025, F₂, 7, 4) (dual of [(1025, 7), 7155, 5]-NRT-code), using
      - appending 3 arbitrary columns [i] based on linear OOA(2²⁰, 1025, F₂, 4, 4) (dual of [(1025, 4), 4080, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2²⁰, 1025, F₂, 3, 4) (dual of [(1025, 3), 3055, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (16, 20, 1025)-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]

(135−8, 135, large)-Net over F₂ — Digital

Digital (127, 135, large)-net over F₂, using

2¹⁸ times duplication [i] based on digital (109, 117, large)-net over F₂, using
- t-expansion [i] based on digital (108, 117, large)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹¹⁷, large, F₂, 9, 9), using
    - appending kth column [i] based on linear OOA(2¹¹⁷, large, F₂, 8, 9), using
      - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹¹⁷, large, F₂, 3, 9), using
        OOAs constructed from BCH codes [i]

(135−8, 135, large)-Net in Base 2 — Upper bound on s

There is no (127, 135, large)-net in base 2, because

6 times m-reduction [i] would yield (127, 129, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]