MinT - Best Known (191−15, 191, s)-Nets in Base 2

Best Known (191−15, 191, s)-Nets in Base 2

(191−15, 191, 1198881)-Net over F₂ — Constructive and digital

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]

(191−15, 191, 1928564)-Net over F₂ — Digital

Digital (176, 191, 1928564)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁹¹, 1928564, F₂, 4, 15) (dual of [(1928564, 4), 7714065, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁹¹, 2097406, F₂, 4, 15) (dual of [(2097406, 4), 8389433, 16]-NRT-code), using
  - OOA 2-folding [i] based on linear OOA(2¹⁹¹, 4194812, F₂, 2, 15) (dual of [(4194812, 2), 8389433, 16]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(2²⁹, 511, F₂, 2, 7) (dual of [(511, 2), 993, 8]-NRT-code), using
        OOAs constructed from BCH codes [i]
      2. 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]

(191−15, 191, large)-Net in Base 2 — Upper bound on s

There is no (176, 191, large)-net in base 2, because

13 times m-reduction [i] would yield (176, 178, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]