MinT - Best Known (123, 123+13, s)-Nets in Base 3

Best Known (123, 123+13, s)-Nets in Base 3

(123, 123+13, 1398142)-Net over F₃ — Constructive and digital

Digital (123, 136, 1398142)-net over F₃, using

3² times duplication [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, 123+13, 4194350)-Net over F₃ — Digital

Digital (123, 136, 4194350)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹³⁶, 4194350, F₃, 2, 13) (dual of [(4194350, 2), 8388564, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁵, 49, F₃, 2, 6) (dual of [(49, 2), 83, 7]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵, 49, F₃, 6) (dual of [49, 34, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵, 85, F₃, 6) (dual of [85, 70, 7]-code), using
        codes constructed by inverting constructionÂ Y₁ [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]

(123, 123+13, large)-Net in Base 3 — Upper bound on s

There is no (123, 136, large)-net in base 3, because

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