MinT - Best Known (191, 207, s)-Nets in Base 2

Best Known (191, 207, s)-Nets in Base 2

(191, 207, 1048614)-Net over F₂ — Constructive and digital

Digital (191, 207, 1048614)-net over F₂, using

2¹ times duplication [i] based on digital (190, 206, 1048614)-net over F₂, using
- t-expansion [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, 207, 1763527)-Net over F₂ — Digital

Digital (191, 207, 1763527)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁰⁷, 1763527, F₂, 4, 16) (dual of [(1763527, 4), 7053901, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁰⁷, 2097190, F₂, 4, 16) (dual of [(2097190, 4), 8388553, 17]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2²¹, 39, F₂, 4, 8) (dual of [(39, 4), 135, 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(2¹⁸⁶, 2097151, F₂, 4, 16) (dual of [(2097151, 4), 8388418, 17]-NRT-code), using
      - OOA 2-folding [i] based on linear OOA(2¹⁸⁶, 4194302, F₂, 2, 16) (dual of [(4194302, 2), 8388418, 17]-NRT-code), using
        1 times NRT-code embedding in larger space [i] based on linear OOA(2¹⁸⁴, 4194301, F₂, 2, 16) (dual of [(4194301, 2), 8388418, 17]-NRT-code), using
        OOA 2-folding [i] based on linear OA(2¹⁸⁴, 8388602, F₂, 16) (dual of [8388602, 8388418, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹⁸⁴, large, F₂, 16) (dual of [large, large−184, 17]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(191, 207, large)-Net in Base 2 — Upper bound on s

There is no (191, 207, large)-net in base 2, because

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