MinT - Best Known (191, 208, s)-Nets in Base 2

Best Known (191, 208, s)-Nets in Base 2

(191, 208, 1048614)-Net over F₂ — Constructive and digital

Digital (191, 208, 1048614)-net over F₂, using

2² times duplication [i] based on digital (189, 206, 1048614)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 21, 39)-net over F₂, using
    - linear code embedding found in a computer search [i]
  2. digital (168, 185, 1048575)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2¹⁸⁵, 1048575, F₂, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
      - OOA 8-folding and stacking with additional row [i] based on linear OA(2¹⁸⁵, 8388601, F₂, 17) (dual of [8388601, 8388416, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹⁸⁵, large, F₂, 17) (dual of [large, large−185, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(191, 208, 1677760)-Net over F₂ — Digital

Digital (191, 208, 1677760)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁰⁸, 1677760, F₂, 5, 17) (dual of [(1677760, 5), 8388592, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²³, 40, F₂, 5, 8) (dual of [(40, 5), 177, 9]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³, 40, F₂, 2, 8) (dual of [(40, 2), 57, 9]-NRT-code), using
      - 1 times NRT-code embedding in larger space [i] based on linear OOA(2²¹, 39, F₂, 2, 8) (dual of [(39, 2), 57, 9]-NRT-code), using
        extracting embedded OOA [i] based on digital (13, 21, 39)-net over F₂, using
        linear code embedding found in a computer search [i]
  2. linear OOA(2¹⁸⁵, 1677720, F₂, 5, 17) (dual of [(1677720, 5), 8388415, 18]-NRT-code), using
    - OOA 5-folding [i] based on linear OA(2¹⁸⁵, 8388600, F₂, 17) (dual of [8388600, 8388415, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁵, large, F₂, 17) (dual of [large, large−185, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(191, 208, large)-Net in Base 2 — Upper bound on s

There is no (191, 208, large)-net in base 2, because

15 times m-reduction [i] would yield (191, 193, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]