MinT - Best Known (256−29, 256, s)-Nets in Base 4

Best Known (256−29, 256, s)-Nets in Base 4

Digital (227, 256, 599185)-net over F₄, using

4³ times duplication [i] based on digital (224, 253, 599185)-net over F₄, using
- net defined by OOA [i] based on linear OOA(4²⁵³, 599185, F₄, 29, 29) (dual of [(599185, 29), 17376112, 30]-NRT-code), using
  - OOA 14-folding and stacking with additional row [i] based on linear OA(4²⁵³, 8388591, F₄, 29) (dual of [8388591, 8388338, 30]-code), using
    - discarding factors / shortening the dual code based on linear OA(4²⁵³, large, F₄, 29) (dual of [large, large−253, 30]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

Digital (227, 256, 2796202)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4²⁵⁶, 2796202, F₄, 3, 29) (dual of [(2796202, 3), 8388350, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4²⁵³, 2796201, F₄, 3, 29) (dual of [(2796201, 3), 8388350, 30]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(4²⁵³, large, F₄, 29) (dual of [large, large−253, 30]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

There is no (227, 256, large)-net in base 4, because

27 times m-reduction [i] would yield (227, 229, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]