MinT - Best Known (169, 169+17, s)-Nets in Base 3

Best Known (169, 169+17, s)-Nets in Base 3

(169, 169+17, 1048615)-Net over F₃ — Constructive and digital

Digital (169, 186, 1048615)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (12, 20, 40)-net over F₃, using
  - trace code for nets [i] based on digital (2, 10, 20)-net over F₉, using
    - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
        a function field by GarcÃa and Quoos [i]
2. digital (149, 166, 1048575)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶⁶, 1048575, F₃, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(3¹⁶⁶, 8388601, F₃, 17) (dual of [8388601, 8388435, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶⁶, large, F₃, 17) (dual of [large, large−166, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(169, 169+17, 4194346)-Net over F₃ — Digital

Digital (169, 186, 4194346)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁸⁶, 4194346, F₃, 2, 17) (dual of [(4194346, 2), 8388506, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3²⁰, 45, F₃, 2, 8) (dual of [(45, 2), 70, 9]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰, 45, F₃, 8) (dual of [45, 25, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²⁰, 46, F₃, 8) (dual of [46, 26, 9]-code), using
        (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
        linear OA(3³, 11, F₃, 2) (dual of [11, 8, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]
        linear OA(3⁵, 11, F₃, 4) (dual of [11, 6, 5]-code), using
        Golay code G(3) [i]
        linear OA(3¹², 24, F₃, 8) (dual of [24, 12, 9]-code), using
        extended quadratic residue code Q_e(24,3) [i]
  2. linear OOA(3¹⁶⁶, 4194301, F₃, 2, 17) (dual of [(4194301, 2), 8388436, 18]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁶⁶, 8388602, F₃, 17) (dual of [8388602, 8388436, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶⁶, large, F₃, 17) (dual of [large, large−166, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(169, 169+17, large)-Net in Base 3 — Upper bound on s

There is no (169, 186, large)-net in base 3, because

15 times m-reduction [i] would yield (169, 171, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (169, 169+17, s)-Nets in Base 3

(169, 169+17, 1048615)-Net over F3 — Constructive and digital

(169, 169+17, 4194346)-Net over F3 — Digital

(169, 169+17, large)-Net in Base 3 — Upper bound on s

(169, 169+17, 1048615)-Net over F₃ — Constructive and digital

(169, 169+17, 4194346)-Net over F₃ — Digital