MinT - Best Known (193−33, 193, s)-Nets in Base 2

Best Known (193−33, 193, s)-Nets in Base 2

Digital (160, 193, 380)-net over F₂, using

2³ times duplication [i] based on digital (157, 190, 380)-net over F₂, using
- trace code for nets [i] based on digital (5, 38, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

Digital (160, 193, 1053)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁹³, 1053, F₂, 3, 33) (dual of [(1053, 3), 2966, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁹³, 1365, F₂, 3, 33) (dual of [(1365, 3), 3902, 34]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(2¹⁹³, 4095, F₂, 33) (dual of [4095, 3902, 34]-code), using
    - discarding factors / shortening the dual code based on linear OA(2¹⁹³, 4096, F₂, 33) (dual of [4096, 3903, 34]-code), using
      - an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]

There is no (160, 193, 27832)-net in base 2, because

1 times m-reduction [i] would yield (160, 192, 27832)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6279 911110 636967 496416 553280 438101 033719 568229 240541 048995 > 2¹⁹² [i]