MinT - Best Known (137−13, 137, s)-Nets in Base 3

Best Known (137−13, 137, s)-Nets in Base 3

(137−13, 137, 1398146)-Net over F₃ — Constructive and digital

Digital (124, 137, 1398146)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (10, 16, 46)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶, 46, F₃, 6, 6) (dual of [(46, 6), 260, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3¹⁶, 46, F₃, 5, 6) (dual of [(46, 5), 214, 7]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(3³, 4, F₃, 5, 3) (dual of [(4, 5), 17, 4]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(5;17,3) [i]
        linear OOA(3¹³, 42, F₃, 5, 6) (dual of [(42, 5), 197, 7]-NRT-code), using
        extracting embedded OOA [i] based on 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]

(137−13, 137, 4194366)-Net over F₃ — Digital

Digital (124, 137, 4194366)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹³⁷, 4194366, F₃, 2, 13) (dual of [(4194366, 2), 8388595, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁶, 65, F₃, 2, 6) (dual of [(65, 2), 114, 7]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶, 65, F₃, 6) (dual of [65, 49, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶, 80, F₃, 6) (dual of [80, 64, 7]-code), using
        the primitive narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [1,5], 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]

(137−13, 137, large)-Net in Base 3 — Upper bound on s

There is no (124, 137, large)-net in base 3, because

11 times m-reduction [i] would yield (124, 126, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]