MinT - Best Known (112, 112+11, s)-Nets in Base 4

Best Known (112, 112+11, s)-Nets in Base 4

Digital (112, 123, 5033160)-net over F₄, using

trace code for nets [i] based on digital (30, 41, 1677720)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
  - OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
    - discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

Digital (112, 123, large)-net over F₄, using

t-expansion [i] based on digital (111, 123, large)-net over F₄, using
- 1 times m-reduction [i] based on digital (111, 124, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²⁴, large, F₄, 13) (dual of [large, large−124, 14]-code), using
    - 15 times code embedding in larger space [i] based on linear OA(4¹⁰⁹, large, F₄, 13) (dual of [large, large−109, 14]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

There is no (112, 123, large)-net in base 4, because

9 times m-reduction [i] would yield (112, 114, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]