MinT - Best Known (172, 186, s)-Nets in Base 2

Best Known (172, 186, s)-Nets in Base 2

(172, 186, 1198498)-Net over F₂ — Constructive and digital

Digital (172, 186, 1198498)-net over F₂, using

2¹ times duplication [i] based on digital (171, 185, 1198498)-net over F₂, using
- t-expansion [i] based on digital (170, 185, 1198498)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (16, 23, 127)-net over F₂, using
      - nets constructed from net-embeddable 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]

(172, 186, 2097278)-Net over F₂ — Digital

Digital (172, 186, 2097278)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁸⁶, 2097278, F₂, 4, 14) (dual of [(2097278, 4), 8388926, 15]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²³, 127, F₂, 4, 7) (dual of [(127, 4), 485, 8]-NRT-code), using
    - extracting embedded OOA [i] based on digital (16, 23, 127)-net over F₂, using
      - nets constructed from net-embeddable BCH codes [i]
  2. linear OOA(2¹⁶³, 2097151, F₂, 4, 14) (dual of [(2097151, 4), 8388441, 15]-NRT-code), using
    - OOA 2-folding [i] based on linear OOA(2¹⁶³, 4194302, F₂, 2, 14) (dual of [(4194302, 2), 8388441, 15]-NRT-code), using
      - 1 times NRT-code embedding in larger space [i] based on linear OOA(2¹⁶¹, 4194301, F₂, 2, 14) (dual of [(4194301, 2), 8388441, 15]-NRT-code), using
        OOA 2-folding [i] based on linear OA(2¹⁶¹, 8388602, F₂, 14) (dual of [8388602, 8388441, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹⁶¹, large, F₂, 14) (dual of [large, large−161, 15]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(172, 186, large)-Net in Base 2 — Upper bound on s

There is no (172, 186, large)-net in base 2, because

12 times m-reduction [i] would yield (172, 174, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]