MinT - Best Known (75−8, 75, s)-Nets in Base 4

Best Known (75−8, 75, s)-Nets in Base 4

(75−8, 75, 2097150)-Net over F₄ — Constructive and digital

Digital (67, 75, 2097150)-net over F₄, using

4² times duplication [i] based on digital (65, 73, 2097150)-net over F₄, using
- t-expansion [i] based on digital (64, 73, 2097150)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁷³, 2097150, F₄, 9, 9) (dual of [(2097150, 9), 18874277, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(4⁷³, 8388601, F₄, 9) (dual of [8388601, 8388528, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁷³, large, F₄, 9) (dual of [large, large−73, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(75−8, 75, large)-Net over F₄ — Digital

Digital (67, 75, large)-net over F₄, using

4³ times duplication [i] based on digital (64, 72, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷², large, F₄, 8) (dual of [large, large−72, 9]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(75−8, 75, large)-Net in Base 4 — Upper bound on s

There is no (67, 75, large)-net in base 4, because

6 times m-reduction [i] would yield (67, 69, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]