MinT - Best Known (145−9, 145, s)-Nets in Base 2

Best Known (145−9, 145, s)-Nets in Base 2

(145−9, 145, 4210686)-Net over F₂ — Constructive and digital

Digital (136, 145, 4210686)-net over F₂, using

2¹ times duplication [i] based on digital (135, 144, 4210686)-net over F₂, using
- net defined by OOA [i] based on linear OOA(2¹⁴⁴, 4210686, F₂, 9, 9) (dual of [(4210686, 9), 37896030, 10]-NRT-code), using
  - appending kth column [i] based on linear OOA(2¹⁴⁴, 4210686, F₂, 8, 9) (dual of [(4210686, 8), 33685344, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2²⁸, 16385, F₂, 8, 4) (dual of [(16385, 8), 131052, 5]-NRT-code), using
        appending 4 arbitrary columns [i] based on linear OOA(2²⁸, 16385, F₂, 4, 4) (dual of [(16385, 4), 65512, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2²⁸, 16385, F₂, 3, 4) (dual of [(16385, 3), 49127, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (24, 28, 16385)-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]

(145−9, 145, large)-Net over F₂ — Digital

Digital (136, 145, large)-net over F₂, using

2⁵ times duplication [i] based on digital (131, 140, large)-net over F₂, using
- t-expansion [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]

(145−9, 145, large)-Net in Base 2 — Upper bound on s

There is no (136, 145, large)-net in base 2, because

7 times m-reduction [i] would yield (136, 138, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]