MinT - Best Known (247−15, 247, s)-Nets in Base 2

Best Known (247−15, 247, s)-Nets in Base 2

(247−15, 247, 2396866)-Net over F₂ — Constructive and digital

Digital (232, 247, 2396866)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (58, 65, 2097150)-net over F₂, using
  - net defined by OOA [i] based on linear OOA(2⁶⁵, 2097150, F₂, 7, 7) (dual of [(2097150, 7), 14679985, 8]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(2⁶⁵, 2097151, F₂, 3, 7) (dual of [(2097151, 3), 6291388, 8]-NRT-code), using
      - OOAs constructed from BCH codes [i]
2. digital (167, 182, 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, 162, 1198371)-net over F₂, using
      - net defined by OOA [i] based on linear OOA(2¹⁶², 1198371, F₂, 15, 15) (dual of [(1198371, 15), 17975403, 16]-NRT-code), using
        OOA 7-folding and stacking with additional row [i] based on linear OA(2¹⁶², 8388598, F₂, 15) (dual of [8388598, 8388436, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹⁶², large, F₂, 15) (dual of [large, large−162, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(247−15, 247, 7402689)-Net over F₂ — Digital

Digital (232, 247, 7402689)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴⁷, 7402689, F₂, 2, 15) (dual of [(7402689, 2), 14805131, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁴⁷, large, F₂, 2, 15), using
  - 2¹ times duplication [i] based on linear OOA(2²⁴⁶, large, F₂, 2, 15), using
    - 8 times NRT-code embedding in larger space [i] based on linear OOA(2²³⁰, 8388602, F₂, 2, 15) (dual of [(8388602, 2), 16776974, 16]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(2⁶⁸, 4194303, F₂, 2, 7) (dual of [(4194303, 2), 8388538, 8]-NRT-code), using
        OOAs constructed from BCH codes [i]
        linear OOA(2¹⁶², 4194301, F₂, 2, 15) (dual of [(4194301, 2), 8388440, 16]-NRT-code), using
        OOA 2-folding [i] based on linear OA(2¹⁶², 8388602, F₂, 15) (dual of [8388602, 8388440, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹⁶², large, F₂, 15) (dual of [large, large−162, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 8388607Â = 2²³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(247−15, 247, large)-Net in Base 2 — Upper bound on s

There is no (232, 247, large)-net in base 2, because

13 times m-reduction [i] would yield (232, 234, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]

Best Known (247−15, 247, s)-Nets in Base 2

(247−15, 247, 2396866)-Net over F2 — Constructive and digital

(247−15, 247, 7402689)-Net over F2 — Digital

(247−15, 247, large)-Net in Base 2 — Upper bound on s

(247−15, 247, 2396866)-Net over F₂ — Constructive and digital

(247−15, 247, 7402689)-Net over F₂ — Digital