MinT - Best Known (162, 175, s)-Nets in Base 2

Best Known (162, 175, s)-Nets in Base 2

(162, 175, 1400147)-Net over F₂ — Constructive and digital

Digital (162, 175, 1400147)-net over F₂, using

2² times duplication [i] based on digital (160, 173, 1400147)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (28, 34, 2047)-net over F₂, using
    - 1 times m-reduction [i] based on digital (28, 35, 2047)-net over F₂, using
      - nets constructed from net-embeddable 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]

(162, 175, 2348372)-Net over F₂ — Digital

Digital (162, 175, 2348372)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁷⁵, 2348372, F₂, 3, 13) (dual of [(2348372, 3), 7044941, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁷⁵, 2798248, F₂, 3, 13) (dual of [(2798248, 3), 8394569, 14]-NRT-code), using
  - 2² times duplication [i] based on linear OOA(2¹⁷³, 2798248, F₂, 3, 13) (dual of [(2798248, 3), 8394571, 14]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2³⁴, 2047, F₂, 3, 6) (dual of [(2047, 3), 6107, 7]-NRT-code), using
        extracting embedded OOA [i] based on digital (28, 34, 2047)-net over F₂, using
        1 times m-reduction [i] based on digital (28, 35, 2047)-net over F₂, using
        nets constructed from net-embeddable 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]

(162, 175, large)-Net in Base 2 — Upper bound on s

There is no (162, 175, large)-net in base 2, because

11 times m-reduction [i] would yield (162, 164, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]