MinT - Best Known (67, 67+9, s)-Nets in Base 4

Best Known (67, 67+9, s)-Nets in Base 4

(67, 67+9, 2097150)-Net over F₄ — Constructive and digital

Digital (67, 76, 2097150)-net over F₄, using

4³ times duplication [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]

(67, 67+9, 4194302)-Net over F₄ — Digital

Digital (67, 76, 4194302)-net over F₄, using

4¹ times duplication [i] based on digital (66, 75, 4194302)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4⁷⁵, 4194302, F₄, 2, 9) (dual of [(4194302, 2), 8388529, 10]-NRT-code), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(4⁷³, 4194301, F₄, 2, 9) (dual of [(4194301, 2), 8388529, 10]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(4⁷³, 8388602, F₄, 9) (dual of [8388602, 8388529, 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]

(67, 67+9, large)-Net in Base 4 — Upper bound on s

There is no (67, 76, large)-net in base 4, because

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