MinT - Best Known (162−13, 162, s)-Nets in Base 2

Best Known (162−13, 162, s)-Nets in Base 2

(162−13, 162, 1398227)-Net over F₂ — Constructive and digital

Digital (149, 162, 1398227)-net over F₂, using

2² times duplication [i] based on digital (147, 160, 1398227)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 21, 127)-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]

(162−13, 162, 2097278)-Net over F₂ — Digital

Digital (149, 162, 2097278)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁶², 2097278, F₂, 4, 13) (dual of [(2097278, 4), 8388950, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²¹, 127, F₂, 4, 6) (dual of [(127, 4), 487, 7]-NRT-code), using
    - extracting embedded OOA [i] based on digital (15, 21, 127)-net over F₂, using
      - linear code embedding found in a computer search [i]
  2. linear OOA(2¹⁴¹, 2097151, F₂, 4, 13) (dual of [(2097151, 4), 8388463, 14]-NRT-code), using
    - OOA 2-folding [i] based on linear OOA(2¹⁴¹, 4194302, F₂, 2, 13) (dual of [(4194302, 2), 8388463, 14]-NRT-code), using
      - 1 times NRT-code embedding in larger space [i] based on 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]

(162−13, 162, large)-Net in Base 2 — Upper bound on s

There is no (149, 162, large)-net in base 2, because

11 times m-reduction [i] would yield (149, 151, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]