MinT - Best Known (187−34, 187, s)-Nets in Base 2

Best Known (187−34, 187, s)-Nets in Base 2

(187−34, 187, 320)-Net over F₂ — Constructive and digital

Digital (153, 187, 320)-net over F₂, using

2² times duplication [i] based on digital (151, 185, 320)-net over F₂, using
- trace code for nets [i] based on digital (3, 37, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]

(187−34, 187, 733)-Net over F₂ — Digital

Digital (153, 187, 733)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁸⁷, 733, F₂, 2, 34) (dual of [(733, 2), 1279, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁸⁷, 1023, F₂, 2, 34) (dual of [(1023, 2), 1859, 35]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(2¹⁸⁷, 2046, F₂, 34) (dual of [2046, 1859, 35]-code), using
    - discarding factors / shortening the dual code based on linear OA(2¹⁸⁷, 2047, F₂, 34) (dual of [2047, 1860, 35]-code), using
      - 1 times truncation [i] based on linear OA(2¹⁸⁸, 2048, F₂, 35) (dual of [2048, 1860, 36]-code), using
        an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 2047Â = 2¹¹âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]

(187−34, 187, 14673)-Net in Base 2 — Upper bound on s

There is no (153, 187, 14674)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 196 252057 262980 641795 530770 967938 455213 785102 021213 356119 > 2¹⁸⁷ [i]