MinT - Best Known (55, 65, s)-Nets in Base 2

Best Known (55, 65, s)-Nets in Base 2

Digital (55, 65, 1638)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁶⁵, 1638, F₂, 10, 10) (dual of [(1638, 10), 16315, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2⁶⁵, 8190, F₂, 10) (dual of [8190, 8125, 11]-code), using
  - discarding factors / shortening the dual code based on linear OA(2⁶⁵, 8191, F₂, 10) (dual of [8191, 8126, 11]-code), using
    - 1 times truncation [i] based on linear OA(2⁶⁶, 8192, F₂, 11) (dual of [8192, 8126, 12]-code), using
      - an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

Digital (55, 65, 2730)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁶⁵, 2730, F₂, 3, 10) (dual of [(2730, 3), 8125, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2⁶⁵, 8190, F₂, 10) (dual of [8190, 8125, 11]-code), using
  - discarding factors / shortening the dual code based on linear OA(2⁶⁵, 8191, F₂, 10) (dual of [8191, 8126, 11]-code), using
    - 1 times truncation [i] based on linear OA(2⁶⁶, 8192, F₂, 11) (dual of [8192, 8126, 12]-code), using
      - an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 8191Â = 2¹³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

There is no (55, 65, 21335)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 36 897269 783902 268268 > 2⁶⁵ [i]