MinT - Best Known (171−16, 171, s)-Nets in Base 3

Best Known (171−16, 171, s)-Nets in Base 3

(171−16, 171, 1048615)-Net over F₃ — Constructive and digital

Digital (155, 171, 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 (135, 151, 1048575)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁵¹, 1048575, F₃, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
    - OA 8-folding and stacking [i] based on linear OA(3¹⁵¹, 8388600, F₃, 16) (dual of [8388600, 8388449, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵¹, large, F₃, 16) (dual of [large, large−151, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(171−16, 171, 4194346)-Net over F₃ — Digital

Digital (155, 171, 4194346)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷¹, 4194346, F₃, 2, 16) (dual of [(4194346, 2), 8388521, 17]-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(3¹⁵¹, 4194301, F₃, 2, 16) (dual of [(4194301, 2), 8388451, 17]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁵¹, 8388602, F₃, 16) (dual of [8388602, 8388451, 17]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵¹, large, F₃, 16) (dual of [large, large−151, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(171−16, 171, large)-Net in Base 3 — Upper bound on s

There is no (155, 171, large)-net in base 3, because

14 times m-reduction [i] would yield (155, 157, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (171−16, 171, s)-Nets in Base 3

(171−16, 171, 1048615)-Net over F3 — Constructive and digital

(171−16, 171, 4194346)-Net over F3 — Digital

(171−16, 171, large)-Net in Base 3 — Upper bound on s

(171−16, 171, 1048615)-Net over F₃ — Constructive and digital

(171−16, 171, 4194346)-Net over F₃ — Digital