MinT - Best Known (123, 135, s)-Nets in Base 3

Best Known (123, 135, s)-Nets in Base 3

(123, 135, 1398142)-Net over F₃ — Constructive and digital

Digital (123, 135, 1398142)-net over F₃, using

3¹ times duplication [i] based on digital (122, 134, 1398142)-net over F₃, using
- t-expansion [i] based on digital (121, 134, 1398142)-net over F₃, using
  - (u,Â u+v)-construction [i] based on
    1. digital (7, 13, 42)-net over F₃, using
      - linear code embedding found in a computer search [i]
    2. 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]

(123, 135, 5602352)-Net over F₃ — Digital

Digital (123, 135, 5602352)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³⁵, 5602352, F₃, 12) (dual of [5602352, 5602217, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³⁵, large, F₃, 12) (dual of [large, large−135, 13]-code), using
  - 15 times code embedding in larger space [i] based on linear OA(3¹²⁰, large, F₃, 12) (dual of [large, large−120, 13]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(123, 135, large)-Net in Base 3 — Upper bound on s

There is no (123, 135, large)-net in base 3, because

10 times m-reduction [i] would yield (123, 125, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]