MinT - Best Known (188−15, 188, s)-Nets in Base 2

Best Known (188−15, 188, s)-Nets in Base 2

(188−15, 188, 1198625)-Net over F₂ — Constructive and digital

Digital (173, 188, 1198625)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 26, 254)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²⁶, 254, F₂, 7, 7) (dual of [(254, 7), 1752, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(2²⁶, 255, F₂, 3, 7) (dual of [(255, 3), 739, 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]

(188−15, 188, 1677975)-Net over F₂ — Digital

Digital (173, 188, 1677975)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁸⁸, 1677975, F₂, 5, 15) (dual of [(1677975, 5), 8389687, 16]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²⁶, 255, F₂, 5, 7) (dual of [(255, 5), 1249, 8]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁶, 255, F₂, 3, 7) (dual of [(255, 3), 739, 8]-NRT-code), using
      - OOAs constructed from BCH codes [i]
  2. linear OOA(2¹⁶², 1677720, F₂, 5, 15) (dual of [(1677720, 5), 8388438, 16]-NRT-code), using
    - OOA 5-folding [i] based on linear OA(2¹⁶², 8388600, F₂, 15) (dual of [8388600, 8388438, 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]

(188−15, 188, large)-Net in Base 2 — Upper bound on s

There is no (173, 188, large)-net in base 2, because

13 times m-reduction [i] would yield (173, 175, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]