MinT - Best Known (230, 230+15, s)-Nets in Base 2

Best Known (230, 230+15, s)-Nets in Base 2

(230, 230+15, 2396824)-Net over F₂ — Constructive and digital

Digital (230, 245, 2396824)-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 (165, 180, 1198412)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (11, 18, 41)-net over F₂, using
      - linear code embedding found in a computer search [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]

(230, 230+15, 6595044)-Net over F₂ — Digital

Digital (230, 245, 6595044)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2²⁴⁵, 6595044, F₂, 2, 15) (dual of [(6595044, 2), 13189843, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2²⁴⁵, large, F₂, 2, 15), using
  - 2¹ times duplication [i] based on linear OOA(2²⁴⁴, large, F₂, 2, 15), using
    - 7 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]

(230, 230+15, large)-Net in Base 2 — Upper bound on s

There is no (230, 245, large)-net in base 2, because

13 times m-reduction [i] would yield (230, 232, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]