MinT - Best Known (123−9, 123, s)-Nets in Base 2

Best Known (123−9, 123, s)-Nets in Base 2

(123−9, 123, 4194312)-Net over F₂ — Constructive and digital

Digital (114, 123, 4194312)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹²³, 4194312, F₂, 9, 9) (dual of [(4194312, 9), 37748685, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2¹²³, 4194312, F₂, 8, 9) (dual of [(4194312, 8), 33554373, 10]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁷, 11, F₂, 8, 4) (dual of [(11, 8), 81, 5]-NRT-code), using
      - appending 4 arbitrary columns [i] based on linear OOA(2⁷, 11, F₂, 4, 4) (dual of [(11, 4), 37, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2⁷, 11, F₂, 3, 4) (dual of [(11, 3), 26, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (3, 7, 11)-net over F₂, using
        hand-crafted matrix by Schmid and Wolf [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]

(123−9, 123, large)-Net over F₂ — Digital

Digital (114, 123, large)-net over F₂, using

2⁶ 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]

(123−9, 123, large)-Net in Base 2 — Upper bound on s

There is no (114, 123, large)-net in base 2, because

7 times m-reduction [i] would yield (114, 116, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]