MinT - Best Known (77−14, 77, s)-Nets in Base 2

Best Known (77−14, 77, s)-Nets in Base 2

Digital (63, 77, 292)-net over F₂, using

net defined by OOA [i] based on linear OOA(2⁷⁷, 292, F₂, 14, 14) (dual of [(292, 14), 4011, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2⁷⁷, 2044, F₂, 14) (dual of [2044, 1967, 15]-code), using
  - discarding factors / shortening the dual code based on linear OA(2⁷⁷, 2047, F₂, 14) (dual of [2047, 1970, 15]-code), using
    - 1 times truncation [i] based on linear OA(2⁷⁸, 2048, F₂, 15) (dual of [2048, 1970, 16]-code), using
      - an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

Digital (63, 77, 682)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁷⁷, 682, F₂, 3, 14) (dual of [(682, 3), 1969, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2⁷⁷, 2046, F₂, 14) (dual of [2046, 1969, 15]-code), using
  - discarding factors / shortening the dual code based on linear OA(2⁷⁷, 2047, F₂, 14) (dual of [2047, 1970, 15]-code), using
    - 1 times truncation [i] based on linear OA(2⁷⁸, 2048, F₂, 15) (dual of [2048, 1970, 16]-code), using
      - an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

There is no (63, 77, 6913)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 151226 184839 500083 124352 > 2⁷⁷ [i]