MinT - Best Known (54, 65, s)-Nets in Base 16

Best Known (54, 65, s)-Nets in Base 16

(54, 65, 3355440)-Net over F₁₆ — Constructive and digital

Digital (54, 65, 3355440)-net over F₁₆, using

16³ times duplication [i] based on digital (51, 62, 3355440)-net over F₁₆, using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256³¹, 1677720, F₂₅₆, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(256³¹, 8388601, F₂₅₆, 11) (dual of [8388601, 8388570, 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]

(54, 65, large)-Net over F₁₆ — Digital

Digital (54, 65, large)-net over F₁₆, using

16⁴ times duplication [i] based on digital (50, 61, large)-net over F₁₆, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16⁶¹, large, F₁₆, 11) (dual of [large, large−61, 12]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 16¹²âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(54, 65, large)-Net in Base 16 — Upper bound on s

There is no (54, 65, large)-net in base 16, because

9 times m-reduction [i] would yield (54, 56, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]