MinT - Best Known (212−12, 212, s)-Nets in Base 2

Best Known (212−12, 212, s)-Nets in Base 2

(212−12, 212, 2796222)-Net over F₂ — Constructive and digital

Digital (200, 212, 2796222)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (58, 64, 2097150)-net over F₂, using
  - 1 times m-reduction [i] based on digital (58, 65, 2097150)-net over F₂, using
    - net defined by OOA [i] based on linear OOA(2⁶⁵, 2097150, F₂, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(2⁶⁵, 2097151, F₂, 3, 7) (dual of [(2097151, 3), 6291388, 8]-NRT-code), using
        OOAs constructed from BCH codes [i]
2. digital (136, 148, 1398111)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (4, 10, 11)-net over F₂, using
      - 1 times m-reduction [i] based on digital (4, 11, 11)-net over F₂, using
        a shift-net [i]
    2. digital (126, 138, 1398100)-net over F₂, using
      - net defined by OOA [i] based on linear OOA(2¹³⁸, 1398100, F₂, 12, 12) (dual of [(1398100, 12), 16777062, 13]-NRT-code), using
        OA 6-folding and stacking [i] based on linear OA(2¹³⁸, 8388600, F₂, 12) (dual of [8388600, 8388462, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹³⁸, large, F₂, 12) (dual of [large, large−138, 13]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(212−12, 212, large)-Net over F₂ — Digital

Digital (200, 212, large)-net over F₂, using

2¹ times duplication [i] based on digital (199, 211, large)-net over F₂, using
- t-expansion [i] based on digital (198, 211, large)-net over F₂, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²¹¹, large, F₂, 2, 13), using
    - 2¹ times duplication [i] based on linear OOA(2²¹⁰, large, F₂, 2, 13), using
      - 2 times NRT-code embedding in larger space [i] based on linear OOA(2²⁰⁶, 8388602, F₂, 2, 13) (dual of [(8388602, 2), 16776998, 14]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(2⁶⁷, 4194303, F₂, 2, 6) (dual of [(4194303, 2), 8388539, 7]-NRT-code), using
        OOAs constructed from BCH codes [i]
        linear OOA(2¹³⁹, 4194301, F₂, 2, 13) (dual of [(4194301, 2), 8388463, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(2¹³⁹, 8388602, F₂, 13) (dual of [8388602, 8388463, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹³⁹, large, F₂, 13) (dual of [large, large−139, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(212−12, 212, large)-Net in Base 2 — Upper bound on s

There is no (200, 212, large)-net in base 2, because

10 times m-reduction [i] would yield (200, 202, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]