MinT - Best Known (256−14, 256, s)-Nets in Base 2

Best Known (256−14, 256, s)-Nets in Base 2

(256−14, 256, 2397762)-Net over F₂ — Constructive and digital

Digital (242, 256, 2397762)-net over F₂, using

t-expansion [i] based on digital (241, 256, 2397762)-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 (176, 191, 1198881)-net over F₂, using
    - (u,Â u+v)-construction [i] based on
      1. digital (22, 29, 510)-net over F₂, using
        net defined by OOA [i] based on linear OOA(2²⁹, 510, F₂, 7, 7) (dual of [(510, 7), 3541, 8]-NRT-code), using
        OOA stacking with additional row [i] based on linear OOA(2²⁹, 511, F₂, 3, 7) (dual of [(511, 3), 1504, 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]

(256−14, 256, large)-Net over F₂ — Digital

Digital (242, 256, large)-net over F₂, using

2⁶ times duplication [i] based on digital (236, 250, large)-net over F₂, using
- t-expansion [i] based on digital (235, 250, large)-net over F₂, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁵⁰, large, F₂, 2, 15), using
    - 10 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]

(256−14, 256, large)-Net in Base 2 — Upper bound on s

There is no (242, 256, large)-net in base 2, because

12 times m-reduction [i] would yield (242, 244, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]

Best Known (256−14, 256, s)-Nets in Base 2

(256−14, 256, 2397762)-Net over F2 — Constructive and digital

(256−14, 256, large)-Net over F2 — Digital

(256−14, 256, large)-Net in Base 2 — Upper bound on s

(256−14, 256, 2397762)-Net over F₂ — Constructive and digital

(256−14, 256, large)-Net over F₂ — Digital