MinT - Best Known (217−17, 217, s)-Nets in Base 2

Best Known (217−17, 217, s)-Nets in Base 2

(217−17, 217, 1048653)-Net over F₂ — Constructive and digital

Digital (200, 217, 1048653)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (24, 32, 78)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2³², 78, F₂, 8, 8) (dual of [(78, 8), 592, 9]-NRT-code), using
    - appending kth column [i] based on linear OOA(2³², 78, F₂, 7, 8) (dual of [(78, 7), 514, 9]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2¹¹, 47, F₂, 7, 4) (dual of [(47, 7), 318, 5]-NRT-code), using
        appending 3 arbitrary columns [i] based on linear OOA(2¹¹, 47, F₂, 4, 4) (dual of [(47, 4), 177, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(2¹¹, 47, F₂, 3, 4) (dual of [(47, 3), 130, 5]-NRT-code), using
        extracting embedded OOA [i] based on digital (7, 11, 47)-net over F₂, using
        linear code embedding found in a computer search [i]
        linear OOA(2²¹, 39, F₂, 7, 8) (dual of [(39, 7), 252, 9]-NRT-code), using
        extracting embedded OOA [i] based on digital (13, 21, 39)-net over F₂, using
        linear code embedding found in a computer search [i]
2. digital (168, 185, 1048575)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2¹⁸⁵, 1048575, F₂, 17, 17) (dual of [(1048575, 17), 17825590, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(2¹⁸⁵, 8388601, F₂, 17) (dual of [8388601, 8388416, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁵, large, F₂, 17) (dual of [large, large−185, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(217−17, 217, 1677848)-Net over F₂ — Digital

Digital (200, 217, 1677848)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²¹⁷, 1677848, F₂, 5, 17) (dual of [(1677848, 5), 8389023, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(2³², 128, F₂, 5, 8) (dual of [(128, 5), 608, 9]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2³², 128, F₂, 2, 8) (dual of [(128, 2), 224, 9]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(2³², 256, F₂, 8) (dual of [256, 224, 9]-code), using
        1 times truncation [i] based on linear OA(2³³, 257, F₂, 9) (dual of [257, 224, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 257Â | 2¹⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
  2. linear OOA(2¹⁸⁵, 1677720, F₂, 5, 17) (dual of [(1677720, 5), 8388415, 18]-NRT-code), using
    - OOA 5-folding [i] based on linear OA(2¹⁸⁵, 8388600, F₂, 17) (dual of [8388600, 8388415, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(2¹⁸⁵, large, F₂, 17) (dual of [large, large−185, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 8388609Â | 2⁴⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(217−17, 217, large)-Net in Base 2 — Upper bound on s

There is no (200, 217, large)-net in base 2, because

15 times m-reduction [i] would yield (200, 202, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]

Best Known (217−17, 217, s)-Nets in Base 2

(217−17, 217, 1048653)-Net over F2 — Constructive and digital

(217−17, 217, 1677848)-Net over F2 — Digital

(217−17, 217, large)-Net in Base 2 — Upper bound on s

(217−17, 217, 1048653)-Net over F₂ — Constructive and digital

(217−17, 217, 1677848)-Net over F₂ — Digital