MinT - Best Known (44, 44+5, s)-Nets in Base 4

Best Known (44, 44+5, s)-Nets in Base 4

(44, 44+5, large)-Net over F₄ — Constructive and digital

Digital (44, 49, large)-net over F₄, using

4¹ times duplication [i] based on digital (43, 48, large)-net over F₄, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 1, 2097151)-net over F₄, using
    - s-reduction based on digital (0, 1, s)-net over F₄ with arbitrarily large s, using
      - construction for strength kÂ =Â 1 [i]
  2. digital (0, 1, 2097151)-net over F₄ (see above)
  3. digital (10, 12, 2097151)-net over F₄, using
    - s-reduction based on digital (10, 12, 5592405)-net over F₄, using
      - kÂ =Â 2 construction [i]
  4. digital (29, 34, 2097151)-net over F₄, using
    - net defined by OOA [i] based on linear OOA(4³⁴, 2097151, F₄, 5, 5) (dual of [(2097151, 5), 10485721, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(4³⁴, 4194303, F₄, 5) (dual of [4194303, 4194269, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(4³⁴, 4194304, F₄, 5) (dual of [4194304, 4194270, 6]-code), using
        an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 4194303Â = 4¹¹âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(44, 44+5, large)-Net in Base 4 — Upper bound on s

There is no (44, 49, large)-net in base 4, because

3 times m-reduction [i] would yield (44, 46, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]