MinT - Best Known (36, 36+9, s)-Nets in Base 2

Best Known (36, 36+9, s)-Nets in Base 2

Digital (36, 45, 512)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁴⁵, 512, F₂, 9, 9) (dual of [(512, 9), 4563, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁴⁵, 512, F₂, 8, 9) (dual of [(512, 8), 4051, 10]-NRT-code), using
  - OOA 4-folding and stacking with additional row [i] based on linear OA(2⁴⁵, 2049, F₂, 9) (dual of [2049, 2004, 10]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 2049Â | 2²²âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

Digital (36, 45, 683)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁴⁵, 683, F₂, 9, 9) (dual of [(683, 9), 6102, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(2⁴⁵, 683, F₂, 8, 9) (dual of [(683, 8), 5419, 10]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁴⁵, 683, F₂, 3, 9) (dual of [(683, 3), 2004, 10]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(2⁴⁵, 2049, F₂, 9) (dual of [2049, 2004, 10]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 2049Â | 2²²âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

There is no (36, 45, 4528)-net in base 2, because

1 times m-reduction [i] would yield (36, 44, 4528)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 17 600385 161629 > 2⁴⁴ [i]