MinT - Best Known (180, 180+14, s)-Nets in Base 2

Best Known (180, 180+14, s)-Nets in Base 2

(180, 180+14, 1199393)-Net over F₂ — Constructive and digital

Digital (180, 194, 1199393)-net over F₂, using

t-expansion [i] based on digital (179, 194, 1199393)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (25, 32, 1022)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2³², 1022, F₂, 7, 7) (dual of [(1022, 7), 7122, 8]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(2³², 1023, F₂, 3, 7) (dual of [(1023, 3), 3037, 8]-NRT-code), using
        OOAs constructed from BCH codes [i]
  2. digital (147, 162, 1198371)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2¹⁶², 1198371, F₂, 15, 15) (dual of [(1198371, 15), 17975403, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(2¹⁶², 8388598, F₂, 15) (dual of [8388598, 8388436, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹⁶², large, F₂, 15) (dual of [large, large−162, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(180, 180+14, 2544760)-Net over F₂ — Digital

Digital (180, 194, 2544760)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁹⁴, 2544760, F₂, 3, 14) (dual of [(2544760, 3), 7634086, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁹⁴, 2797224, F₂, 3, 14) (dual of [(2797224, 3), 8391478, 15]-NRT-code), using
  - strength reduction [i] based on linear OOA(2¹⁹⁴, 2797224, F₂, 3, 15) (dual of [(2797224, 3), 8391478, 16]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2³², 1023, F₂, 3, 7) (dual of [(1023, 3), 3037, 8]-NRT-code), using
        OOAs constructed from BCH codes [i]
      2. linear OOA(2¹⁶², 2796201, F₂, 3, 15) (dual of [(2796201, 3), 8388441, 16]-NRT-code), using
        OOA 3-folding [i] based on linear OA(2¹⁶², large, F₂, 15) (dual of [large, large−162, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(180, 180+14, large)-Net in Base 2 — Upper bound on s

There is no (180, 194, large)-net in base 2, because

12 times m-reduction [i] would yield (180, 182, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]