MinT - Best Known (142, 155, s)-Nets in Base 2

Best Known (142, 155, s)-Nets in Base 2

(142, 155, 1398142)-Net over F₂ — Constructive and digital

Digital (142, 155, 1398142)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (10, 16, 42)-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]

(142, 155, 1809792)-Net over F₂ — Digital

Digital (142, 155, 1809792)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁵⁵, 1809792, F₂, 4, 13) (dual of [(1809792, 4), 7239013, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁵⁵, 2097171, F₂, 4, 13) (dual of [(2097171, 4), 8388529, 14]-NRT-code), using
  - OOA 2-folding [i] based on linear OOA(2¹⁵⁵, 4194342, F₂, 2, 13) (dual of [(4194342, 2), 8388529, 14]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(2¹⁵⁵, 4194343, F₂, 2, 13) (dual of [(4194343, 2), 8388531, 14]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2¹⁶, 42, F₂, 2, 6) (dual of [(42, 2), 68, 7]-NRT-code), using
        extracting embedded OOA [i] based on digital (10, 16, 42)-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]

(142, 155, large)-Net in Base 2 — Upper bound on s

There is no (142, 155, large)-net in base 2, because

11 times m-reduction [i] would yield (142, 144, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]