MinT - Best Known (125−9, 125, s)-Nets in Base 2

Best Known (125−9, 125, s)-Nets in Base 2

(125−9, 125, 4194324)-Net over F₂ — Constructive and digital

Digital (116, 125, 4194324)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹²⁵, 4194324, F₂, 9, 9) (dual of [(4194324, 9), 37748791, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹²⁵, 4194324, F₂, 8, 9) (dual of [(4194324, 8), 33554467, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁹, 23, F₂, 8, 4) (dual of [(23, 8), 175, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2⁹, 23, F₂, 4, 4) (dual of [(23, 4), 83, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁹, 23, F₂, 3, 4) (dual of [(23, 3), 60, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (5, 9, 23)-net over F₂, using
        linear code embedding found in a computer search [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]

(125−9, 125, large)-Net over F₂ — Digital

Digital (116, 125, large)-net over F₂, using

2⁸ 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]

(125−9, 125, large)-Net in Base 2 — Upper bound on s

There is no (116, 125, large)-net in base 2, because

7 times m-reduction [i] would yield (116, 118, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]