MinT - Best Known (193−13, 193, s)-Nets in Base 2

Best Known (193−13, 193, s)-Nets in Base 2

(193−13, 193, 1529171)-Net over F₂ — Constructive and digital

Digital (180, 193, 1529171)-net over F₂, using

2² times duplication [i] based on digital (178, 191, 1529171)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (46, 52, 131071)-net over F₂, using
    - 1 times m-reduction [i] based on digital (46, 53, 131071)-net over F₂, using
      - nets constructed from net-embeddable 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]

(193−13, 193, 2927273)-Net over F₂ — Digital

Digital (180, 193, 2927273)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁹³, 2927273, F₂, 3, 13) (dual of [(2927273, 3), 8781626, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2⁵⁴, 131072, F₂, 3, 6) (dual of [(131072, 3), 393162, 7]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁵⁴, 131072, F₂, 2, 6) (dual of [(131072, 2), 262090, 7]-NRT-code), using
      - 1 times NRT-code embedding in larger space [i] based on linear OOA(2⁵², 131071, F₂, 2, 6) (dual of [(131071, 2), 262090, 7]-NRT-code), using
        extracting embedded OOA [i] based on digital (46, 52, 131071)-net over F₂, using
        1 times m-reduction [i] based on digital (46, 53, 131071)-net over F₂, using
        nets constructed from net-embeddable BCH codes [i]
  2. linear OOA(2¹³⁹, 2796201, F₂, 3, 13) (dual of [(2796201, 3), 8388464, 14]-NRT-code), using
    - OOA 3-folding [i] 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]

(193−13, 193, large)-Net in Base 2 — Upper bound on s

There is no (180, 193, large)-net in base 2, because

11 times m-reduction [i] would yield (180, 182, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]