MinT - Best Known (127−14, 127, s)-Nets in Base 4

Best Known (127−14, 127, s)-Nets in Base 4

Digital (113, 127, 1198371)-net over F₄, using

4⁶ times duplication [i] based on digital (107, 121, 1198371)-net over F₄, using
- net defined by OOA [i] based on linear OOA(4¹²¹, 1198371, F₄, 14, 14) (dual of [(1198371, 14), 16777073, 15]-NRT-code), using
  - OA 7-folding and stacking [i] based on linear OA(4¹²¹, 8388597, F₄, 14) (dual of [8388597, 8388476, 15]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹²¹, large, F₄, 14) (dual of [large, large−121, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

Digital (113, 127, 4194304)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹²⁷, 4194304, F₄, 2, 14) (dual of [(4194304, 2), 8388481, 15]-NRT-code), using
- 3 times NRT-code embedding in larger space [i] based on linear OOA(4¹²¹, 4194301, F₄, 2, 14) (dual of [(4194301, 2), 8388481, 15]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(4¹²¹, 8388602, F₄, 14) (dual of [8388602, 8388481, 15]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹²¹, large, F₄, 14) (dual of [large, large−121, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

There is no (113, 127, large)-net in base 4, because

12 times m-reduction [i] would yield (113, 115, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]