MinT - Best Known (160, 174, s)-Nets in Base 2

Best Known (160, 174, s)-Nets in Base 2

(160, 174, 1198387)-Net over F₂ — Constructive and digital

Digital (160, 174, 1198387)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 13, 16)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (147, 161, 1198371)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁶¹, 1198371, F₂, 14, 14) (dual of [(1198371, 14), 16777033, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(2¹⁶¹, 8388597, F₂, 14) (dual of [8388597, 8388436, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁶¹, large, F₂, 14) (dual of [large, large−161, 15]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(160, 174, 2013126)-Net over F₂ — Digital

Digital (160, 174, 2013126)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁷⁴, 2013126, F₂, 4, 14) (dual of [(2013126, 4), 8052330, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁷⁴, 2097158, F₂, 4, 14) (dual of [(2097158, 4), 8388458, 15]-NRT-code), using
  - OOA 2-folding [i] based on linear OOA(2¹⁷⁴, 4194316, F₂, 2, 14) (dual of [(4194316, 2), 8388458, 15]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(2¹⁷⁴, 4194317, F₂, 2, 14) (dual of [(4194317, 2), 8388460, 15]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2¹³, 16, F₂, 2, 7) (dual of [(16, 2), 19, 8]-NRT-code), using
        extracting embedded OOA [i] based on digital (6, 13, 16)-net over F₂, using
        linear code embedding found in a computer search [i]
        linear OOA(2¹⁶¹, 4194301, F₂, 2, 14) (dual of [(4194301, 2), 8388441, 15]-NRT-code), using
        OOA 2-folding [i] based on linear OA(2¹⁶¹, 8388602, F₂, 14) (dual of [8388602, 8388441, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹⁶¹, large, F₂, 14) (dual of [large, large−161, 15]-code), using
        the primitive narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(160, 174, large)-Net in Base 2 — Upper bound on s

There is no (160, 174, large)-net in base 2, because

12 times m-reduction [i] would yield (160, 162, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]