MinT - Best Known (251−10, 251, s)-Nets in Base 2

Best Known (251−10, 251, s)-Nets in Base 2

(251−10, 251, large)-Net over F₂ — Constructive and digital

Digital (241, 251, large)-net over F₂, using

2³ times duplication [i] based on digital (238, 248, large)-net over F₂, using
- t-expansion [i] based on digital (237, 248, large)-net over F₂, using
  - trace code for nets [i] based on digital (51, 62, 2796200)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁶², 2796200, F₁₆, 14, 11) (dual of [(2796200, 14), 39146738, 12]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁶², 8388601, F₁₆, 2, 11) (dual of [(8388601, 2), 16777140, 12]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁶², 8388602, F₁₆, 2, 11) (dual of [(8388602, 2), 16777142, 12]-NRT-code), using
        trace code [i] based on linear OOA(256³¹, 4194301, F₂₅₆, 2, 11) (dual of [(4194301, 2), 8388571, 12]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³¹, 8388602, F₂₅₆, 11) (dual of [8388602, 8388571, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(256³¹, large, F₂₅₆, 11) (dual of [large, large−31, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(251−10, 251, large)-Net in Base 2 — Upper bound on s

There is no (241, 251, large)-net in base 2, because

8 times m-reduction [i] would yield (241, 243, large)-net in base 2, but
- mutually orthogonal hypercube bound [i]