MinT - Best Known (182−12, 182, s)-Nets in Base 2

Best Known (182−12, 182, s)-Nets in Base 2

(182−12, 182, 1414482)-Net over F₂ — Constructive and digital

Digital (170, 182, 1414482)-net over F₂, using

t-expansion [i] based on digital (169, 182, 1414482)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (37, 43, 16382)-net over F₂, using
    - 1 times m-reduction [i] based on digital (37, 44, 16382)-net over F₂, using
      - net defined by OOA [i] based on linear OOA(2⁴⁴, 16382, F₂, 7, 7) (dual of [(16382, 7), 114630, 8]-NRT-code), using
        OOA stacking with additional row [i] based on linear OOA(2⁴⁴, 16383, F₂, 3, 7) (dual of [(16383, 3), 49105, 8]-NRT-code), using
        OOAs constructed from BCH codes [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]

(182−12, 182, 4210684)-Net over F₂ — Digital

Digital (170, 182, 4210684)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁸², 4210684, F₂, 2, 12) (dual of [(4210684, 2), 8421186, 13]-NRT-code), using
- strength reduction [i] based on linear OOA(2¹⁸², 4210684, F₂, 2, 13) (dual of [(4210684, 2), 8421186, 14]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(2⁴³, 16383, F₂, 2, 6) (dual of [(16383, 2), 32723, 7]-NRT-code), using
      - OOAs constructed from BCH codes [i]
    2. 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]

(182−12, 182, large)-Net in Base 2 — Upper bound on s

There is no (170, 182, large)-net in base 2, because

10 times m-reduction [i] would yield (170, 172, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]