MinT - Best Known (173−14, 173, s)-Nets in Base 2

Best Known (173−14, 173, s)-Nets in Base 2

(173−14, 173, 1198384)-Net over F₂ — Constructive and digital

Digital (159, 173, 1198384)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 12, 13)-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]

(173−14, 173, 1863901)-Net over F₂ — Digital

Digital (159, 173, 1863901)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁷³, 1863901, F₂, 4, 14) (dual of [(1863901, 4), 7455431, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁷³, 2097157, F₂, 4, 14) (dual of [(2097157, 4), 8388455, 15]-NRT-code), using
  - OOA 2-folding [i] based on linear OOA(2¹⁷³, 4194314, F₂, 2, 14) (dual of [(4194314, 2), 8388455, 15]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2¹², 13, F₂, 2, 7) (dual of [(13, 2), 14, 8]-NRT-code), using
        extracting embedded OOA [i] based on digital (5, 12, 13)-net over F₂, using
        linear code embedding found in a computer search [i]
      2. 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]

(173−14, 173, large)-Net in Base 2 — Upper bound on s

There is no (159, 173, large)-net in base 2, because

12 times m-reduction [i] would yield (159, 161, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]