MinT - Best Known (156−13, 156, s)-Nets in Base 2

Best Known (156−13, 156, s)-Nets in Base 2

(156−13, 156, 1398148)-Net over F₂ — Constructive and digital

Digital (143, 156, 1398148)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 17, 48)-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]

(156−13, 156, 1973593)-Net over F₂ — Digital

Digital (143, 156, 1973593)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁵⁶, 1973593, F₂, 4, 13) (dual of [(1973593, 4), 7894216, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁵⁶, 2097174, F₂, 4, 13) (dual of [(2097174, 4), 8388540, 14]-NRT-code), using
  - OOA 2-folding [i] based on linear OOA(2¹⁵⁶, 4194348, F₂, 2, 13) (dual of [(4194348, 2), 8388540, 14]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(2¹⁵⁶, 4194349, F₂, 2, 13) (dual of [(4194349, 2), 8388542, 14]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2¹⁷, 48, F₂, 2, 6) (dual of [(48, 2), 79, 7]-NRT-code), using
        extracting embedded OOA [i] based on digital (11, 17, 48)-net over F₂, using
        linear code embedding found in a computer search [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]

(156−13, 156, large)-Net in Base 2 — Upper bound on s

There is no (143, 156, large)-net in base 2, because

11 times m-reduction [i] would yield (143, 145, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]