MinT - Best Known (124−13, 124, s)-Nets in Base 3

Best Known (124−13, 124, s)-Nets in Base 3

Digital (111, 124, 1398100)-net over F₃, using

3³ times duplication [i] based on digital (108, 121, 1398100)-net over F₃, using
- net defined by OOA [i] based on linear OOA(3¹²¹, 1398100, F₃, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
  - OOA 6-folding and stacking with additional row [i] based on linear OA(3¹²¹, 8388601, F₃, 13) (dual of [8388601, 8388480, 14]-code), using
    - discarding factors / shortening the dual code based on linear OA(3¹²¹, large, F₃, 13) (dual of [large, large−121, 14]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

Digital (111, 124, 2796202)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹²⁴, 2796202, F₃, 3, 13) (dual of [(2796202, 3), 8388482, 14]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3¹²¹, 2796201, F₃, 3, 13) (dual of [(2796201, 3), 8388482, 14]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3¹²¹, large, F₃, 13) (dual of [large, large−121, 14]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

There is no (111, 124, large)-net in base 3, because

11 times m-reduction [i] would yield (111, 113, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]