MinT - Best Known (140, 140+28, s)-Nets in Base 2

Best Known (140, 140+28, s)-Nets in Base 2

(140, 140+28, 390)-Net over F₂ — Constructive and digital

Digital (140, 168, 390)-net over F₂, using

trace code for nets [i] based on digital (0, 28, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(140, 140+28, 1116)-Net over F₂ — Digital

Digital (140, 168, 1116)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁶⁸, 1116, F₂, 3, 28) (dual of [(1116, 3), 3180, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁶⁸, 1365, F₂, 3, 28) (dual of [(1365, 3), 3927, 29]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(2¹⁶⁸, 4095, F₂, 28) (dual of [4095, 3927, 29]-code), using
    - discarding factors / shortening the dual code based on linear OA(2¹⁶⁸, 4096, F₂, 28) (dual of [4096, 3928, 29]-code), using
      - 1 times truncation [i] based on linear OA(2¹⁶⁹, 4097, F₂, 29) (dual of [4097, 3928, 30]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 4097Â | 2²⁴âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]

(140, 140+28, 24743)-Net in Base 2 — Upper bound on s

There is no (140, 168, 24744)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 374 286065 424554 819810 872647 063342 129161 611224 234108 > 2¹⁶⁸ [i]