MinT - Best Known (141−13, 141, s)-Nets in Base 3

Best Known (141−13, 141, s)-Nets in Base 3

(141−13, 141, 1398190)-Net over F₃ — Constructive and digital

Digital (128, 141, 1398190)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (14, 20, 90)-net over F₃, using
  - trace code for nets [i] based on digital (4, 10, 45)-net over F₉, using
    - Niederreiter sequence (Bratleyâ€“Foxâ€“Niederreiter implementation) [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]

(141−13, 141, 4194502)-Net over F₃ — Digital

Digital (128, 141, 4194502)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁴¹, 4194502, F₃, 2, 13) (dual of [(4194502, 2), 8388863, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3²⁰, 201, F₃, 2, 6) (dual of [(201, 2), 382, 7]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰, 201, F₃, 6) (dual of [201, 181, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²⁰, 242, F₃, 6) (dual of [242, 222, 7]-code), using
        the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  2. linear OOA(3¹²¹, 4194301, F₃, 2, 13) (dual of [(4194301, 2), 8388481, 14]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹²¹, 8388602, F₃, 13) (dual of [8388602, 8388481, 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]

(141−13, 141, large)-Net in Base 3 — Upper bound on s

There is no (128, 141, large)-net in base 3, because

11 times m-reduction [i] would yield (128, 130, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]