MinT - Best Known (223−13, 223, s)-Nets in Base 2

Best Known (223−13, 223, s)-Nets in Base 2

(223−13, 223, 2796358)-Net over F₂ — Constructive and digital

Digital (210, 223, 2796358)-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 (146, 159, 1398179)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 20, 79)-net over F₂, using
      - linear code embedding found in a computer search [i]
    2. digital (126, 139, 1398100)-net over F₂, using
      - net defined by OOA [i] based on linear OOA(2¹³⁹, 1398100, F₂, 13, 13) (dual of [(1398100, 13), 18175161, 14]-NRT-code), using
        OOA 6-folding and stacking with additional row [i] based on linear OA(2¹³⁹, 8388601, F₂, 13) (dual of [8388601, 8388462, 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]

(223−13, 223, large)-Net over F₂ — Digital

Digital (210, 223, large)-net over F₂, using

2¹² times duplication [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]

(223−13, 223, large)-Net in Base 2 — Upper bound on s

There is no (210, 223, large)-net in base 2, because

11 times m-reduction [i] would yield (210, 212, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]