MinT - Best Known (188−13, 188, s)-Nets in Base 2

Best Known (188−13, 188, s)-Nets in Base 2

(188−13, 188, 1463634)-Net over F₂ — Constructive and digital

Digital (175, 188, 1463634)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (43, 49, 65534)-net over F₂, using
  - 1 times m-reduction [i] based on digital (43, 50, 65534)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2⁵⁰, 65534, F₂, 7, 7) (dual of [(65534, 7), 458688, 8]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(2⁵⁰, 65535, F₂, 3, 7) (dual of [(65535, 3), 196555, 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]

(188−13, 188, 2861736)-Net over F₂ — Digital

Digital (175, 188, 2861736)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁸⁸, 2861736, F₂, 3, 13) (dual of [(2861736, 3), 8585020, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁴⁹, 65535, F₂, 3, 6) (dual of [(65535, 3), 196556, 7]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁴⁹, 65535, F₂, 2, 6) (dual of [(65535, 2), 131021, 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]

(188−13, 188, large)-Net in Base 2 — Upper bound on s

There is no (175, 188, large)-net in base 2, because

11 times m-reduction [i] would yield (175, 177, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]