MinT - Best Known (137−9, 137, s)-Nets in Base 2

Best Known (137−9, 137, s)-Nets in Base 2

(137−9, 137, 4195326)-Net over F₂ — Constructive and digital

Digital (128, 137, 4195326)-net over F₂, using

2¹ times duplication [i] based on digital (127, 136, 4195326)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2¹³⁶, 4195326, F₂, 9, 9) (dual of [(4195326, 9), 37757798, 10]-NRT-code), using
  - appending kth column [i] based on linear OOA(2¹³⁶, 4195326, F₂, 8, 9) (dual of [(4195326, 8), 33562472, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2²⁰, 1025, F₂, 8, 4) (dual of [(1025, 8), 8180, 5]-NRT-code), using
        appending 4 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(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]

(137−9, 137, large)-Net over F₂ — Digital

Digital (128, 137, large)-net over F₂, using

2²⁰ times duplication [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]

(137−9, 137, large)-Net in Base 2 — Upper bound on s

There is no (128, 137, large)-net in base 2, because

7 times m-reduction [i] would yield (128, 130, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]