MinT - Best Known (161−12, 161, s)-Nets in Base 2

Best Known (161−12, 161, s)-Nets in Base 2

(161−12, 161, 1398227)-Net over F₂ — Constructive and digital

Digital (149, 161, 1398227)-net over F₂, using

2¹ times duplication [i] based on digital (148, 160, 1398227)-net over F₂, using
- t-expansion [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]

(161−12, 161, 2796329)-Net over F₂ — Digital

Digital (149, 161, 2796329)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁶¹, 2796329, F₂, 3, 12) (dual of [(2796329, 3), 8388826, 13]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²³, 128, F₂, 3, 6) (dual of [(128, 3), 361, 7]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²³, 128, F₂, 2, 6) (dual of [(128, 2), 233, 7]-NRT-code), using
      - 1 times NRT-code embedding in larger space [i] based on linear OOA(2²¹, 127, F₂, 2, 6) (dual of [(127, 2), 233, 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(2¹³⁸, 2796201, F₂, 3, 12) (dual of [(2796201, 3), 8388465, 13]-NRT-code), using
    - OOA 3-folding [i] 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]

(161−12, 161, large)-Net in Base 2 — Upper bound on s

There is no (149, 161, large)-net in base 2, because

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