MinT - Best Known (223−7, 223, s)-Nets in Base 2

Best Known (223−7, 223, s)-Nets in Base 2

(223−7, 223, large)-Net over F₂ — Constructive and digital

Digital (216, 223, large)-net over F₂, using

t-expansion [i] based on digital (214, 223, large)-net over F₂, using
- 1 times m-reduction [i] based on digital (214, 224, large)-net over F₂, using
  - trace code for nets [i] based on digital (46, 56, 2796200)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁵⁶, 2796200, F₁₆, 14, 10) (dual of [(2796200, 14), 39146744, 11]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁵⁶, 8388601, F₁₆, 2, 10) (dual of [(8388601, 2), 16777146, 11]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁵⁶, 8388602, F₁₆, 2, 10) (dual of [(8388602, 2), 16777148, 11]-NRT-code), using
        trace code [i] based on linear OOA(256²⁸, 4194301, F₂₅₆, 2, 10) (dual of [(4194301, 2), 8388574, 11]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256²⁸, 8388602, F₂₅₆, 10) (dual of [8388602, 8388574, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(256²⁸, large, F₂₅₆, 10) (dual of [large, large−28, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(223−7, 223, large)-Net in Base 2 — Upper bound on s

There is no (216, 223, large)-net in base 2, because

5 times m-reduction [i] would yield (216, 218, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]