MinT - Best Known (175−17, 175, s)-Nets in Base 3

Best Known (175−17, 175, s)-Nets in Base 3

(175−17, 175, 1048582)-Net over F₃ — Constructive and digital

Digital (158, 175, 1048582)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 9, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [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]

(175−17, 175, 2796208)-Net over F₃ — Digital

Digital (158, 175, 2796208)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷⁵, 2796208, F₃, 3, 17) (dual of [(2796208, 3), 8388449, 18]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3⁹, 7, F₃, 3, 8) (dual of [(7, 3), 12, 9]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,12P) [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
  2. linear OOA(3¹⁶⁶, 2796201, F₃, 3, 17) (dual of [(2796201, 3), 8388437, 18]-NRT-code), using
    - OOA 3-folding [i] 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]

(175−17, 175, large)-Net in Base 3 — Upper bound on s

There is no (158, 175, large)-net in base 3, because

15 times m-reduction [i] would yield (158, 160, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]