MinT - Best Known (187−13, 187, s)-Nets in Base 2

Best Known (187−13, 187, s)-Nets in Base 2

(187−13, 187, 1430866)-Net over F₂ — Constructive and digital

Digital (174, 187, 1430866)-net over F₂, using

2² times duplication [i] based on digital (172, 185, 1430866)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (40, 46, 32766)-net over F₂, using
    - 1 times m-reduction [i] based on 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 (126, 139, 1398100)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2¹³⁹, 1398100, F₂, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
      - OOA 6-folding and stacking with additional row [i] based on linear OA(2¹³⁹, 8388601, F₂, 13) (dual of [8388601, 8388462, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹³⁹, large, F₂, 13) (dual of [large, large−139, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(187−13, 187, 2828969)-Net over F₂ — Digital

Digital (174, 187, 2828969)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁸⁷, 2828969, F₂, 3, 13) (dual of [(2828969, 3), 8486720, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁴⁸, 32768, F₂, 3, 6) (dual of [(32768, 3), 98256, 7]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁴⁸, 32768, F₂, 2, 6) (dual of [(32768, 2), 65488, 7]-NRT-code), using
      - 1 times NRT-code embedding in larger space [i] based on linear OOA(2⁴⁶, 32767, F₂, 2, 6) (dual of [(32767, 2), 65488, 7]-NRT-code), using
        OOAs constructed from BCH codes [i]
  2. linear OOA(2¹³⁹, 2796201, F₂, 3, 13) (dual of [(2796201, 3), 8388464, 14]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(2¹³⁹, large, F₂, 13) (dual of [large, large−139, 14]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(187−13, 187, large)-Net in Base 2 — Upper bound on s

There is no (174, 187, large)-net in base 2, because

11 times m-reduction [i] would yield (174, 176, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]