MinT - Best Known (211−15, 211, s)-Nets in Base 2

Best Known (211−15, 211, s)-Nets in Base 2

(211−15, 211, 1231137)-Net over F₂ — Constructive and digital

Digital (196, 211, 1231137)-net over F₂, using

2² times duplication [i] based on digital (194, 209, 1231137)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (40, 47, 32766)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2⁴⁷, 32766, F₂, 7, 7) (dual of [(32766, 7), 229315, 8]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(2⁴⁷, 32767, F₂, 3, 7) (dual of [(32767, 3), 98254, 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]

(211−15, 211, 2416657)-Net over F₂ — Digital

Digital (196, 211, 2416657)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²¹¹, 2416657, F₂, 3, 15) (dual of [(2416657, 3), 7249760, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²¹¹, 2828968, F₂, 3, 15) (dual of [(2828968, 3), 8486693, 16]-NRT-code), using
  - 2² times duplication [i] based on linear OOA(2²⁰⁹, 2828968, F₂, 3, 15) (dual of [(2828968, 3), 8486695, 16]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2⁴⁷, 32767, F₂, 3, 7) (dual of [(32767, 3), 98254, 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]

(211−15, 211, large)-Net in Base 2 — Upper bound on s

There is no (196, 211, large)-net in base 2, because

13 times m-reduction [i] would yield (196, 198, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]