MinT - Best Known (168, 168+17, s)-Nets in Base 3

Best Known (168, 168+17, s)-Nets in Base 3

(168, 168+17, 1048607)-Net over F₃ — Constructive and digital

Digital (168, 185, 1048607)-net over F₃, using

3¹ times duplication [i] based on digital (167, 184, 1048607)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 18, 32)-net over F₃, using
    - trace code for nets [i] based on digital (1, 9, 16)-net over F₉, using
      - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, 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]

(168, 168+17, 4194338)-Net over F₃ — Digital

Digital (168, 185, 4194338)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁸⁵, 4194338, F₃, 2, 17) (dual of [(4194338, 2), 8388491, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁹, 37, F₃, 2, 8) (dual of [(37, 2), 55, 9]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹, 37, F₃, 8) (dual of [37, 18, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁹, 48, F₃, 8) (dual of [48, 29, 9]-code), using
        (u,Â u+v)-construction [i] based on
        linear OA(3⁷, 24, F₃, 4) (dual of [24, 17, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁷, 26, F₃, 4) (dual of [26, 19, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 26Â = 3³âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [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]

(168, 168+17, large)-Net in Base 3 — Upper bound on s

There is no (168, 185, large)-net in base 3, because

15 times m-reduction [i] would yield (168, 170, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (168, 168+17, s)-Nets in Base 3

(168, 168+17, 1048607)-Net over F3 — Constructive and digital

(168, 168+17, 4194338)-Net over F3 — Digital

(168, 168+17, large)-Net in Base 3 — Upper bound on s

(168, 168+17, 1048607)-Net over F₃ — Constructive and digital

(168, 168+17, 4194338)-Net over F₃ — Digital