MinT - Best Known (181−14, 181, s)-Nets in Base 2

Best Known (181−14, 181, s)-Nets in Base 2

(181−14, 181, 1198433)-Net over F₂ — Constructive and digital

Digital (167, 181, 1198433)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (13, 20, 62)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2²⁰, 62, F₂, 7, 7) (dual of [(62, 7), 414, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(2²⁰, 63, F₂, 3, 7) (dual of [(63, 3), 169, 8]-NRT-code), using
      - OOAs constructed from BCH codes [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]

(181−14, 181, 2097213)-Net over F₂ — Digital

Digital (167, 181, 2097213)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁸¹, 2097213, F₂, 4, 14) (dual of [(2097213, 4), 8388671, 15]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2²⁰, 63, F₂, 4, 7) (dual of [(63, 4), 232, 8]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁰, 63, F₂, 3, 7) (dual of [(63, 3), 169, 8]-NRT-code), using
      - OOAs constructed from BCH codes [i]
  2. linear OOA(2¹⁶¹, 2097150, F₂, 4, 14) (dual of [(2097150, 4), 8388439, 15]-NRT-code), using
    - OOA 4-folding [i] based on linear OA(2¹⁶¹, 8388600, F₂, 14) (dual of [8388600, 8388439, 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]

(181−14, 181, large)-Net in Base 2 — Upper bound on s

There is no (167, 181, large)-net in base 2, because

12 times m-reduction [i] would yield (167, 169, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]