MinT - Best Known (171−13, 171, s)-Nets in Base 2

Best Known (171−13, 171, s)-Nets in Base 2

(171−13, 171, 1399122)-Net over F₂ — Constructive and digital

Digital (158, 171, 1399122)-net over F₂, using

2¹ times duplication [i] based on digital (157, 170, 1399122)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (25, 31, 1022)-net over F₂, using
    - 1 times m-reduction [i] based on digital (25, 32, 1022)-net over F₂, using
      - net defined by OOA [i] based on linear OOA(2³², 1022, F₂, 7, 7) (dual of [(1022, 7), 7122, 8]-NRT-code), using
        OOA stacking with additional row [i] based on linear OOA(2³², 1023, F₂, 3, 7) (dual of [(1023, 3), 3037, 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]

(171−13, 171, 2098173)-Net over F₂ — Digital

Digital (158, 171, 2098173)-net over F₂, using

2¹ times duplication [i] based on digital (157, 170, 2098173)-net over F₂, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁷⁰, 2098173, F₂, 4, 13) (dual of [(2098173, 4), 8392522, 14]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2³¹, 1023, F₂, 4, 6) (dual of [(1023, 4), 4061, 7]-NRT-code), using
      - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2³¹, 1023, F₂, 2, 6) (dual of [(1023, 2), 2015, 7]-NRT-code), using
        OOAs constructed from BCH codes [i]
    2. linear OOA(2¹³⁹, 2097150, F₂, 4, 13) (dual of [(2097150, 4), 8388461, 14]-NRT-code), using
      - OOA 4-folding [i] based on linear OA(2¹³⁹, 8388600, F₂, 13) (dual of [8388600, 8388461, 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]

(171−13, 171, large)-Net in Base 2 — Upper bound on s

There is no (158, 171, large)-net in base 2, because

11 times m-reduction [i] would yield (158, 160, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]