MinT - Best Known (185, 185+19, s)-Nets in Base 3

Best Known (185, 185+19, s)-Nets in Base 3

(185, 185+19, 932106)-Net over F₃ — Constructive and digital

Digital (185, 204, 932106)-net over F₃, using

3¹ times duplication [i] based on digital (184, 203, 932106)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 22, 40)-net over F₃, using
    - trace code for nets [i] based on digital (2, 11, 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 (162, 181, 932066)-net over F₃, using
    - net defined by OOA [i] based on linear OOA(3¹⁸¹, 932066, F₃, 19, 19) (dual of [(932066, 19), 17709073, 20]-NRT-code), using
      - OOA 9-folding and stacking with additional row [i] based on linear OA(3¹⁸¹, 8388595, F₃, 19) (dual of [8388595, 8388414, 20]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁸¹, large, F₃, 19) (dual of [large, large−181, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(185, 185+19, 3590565)-Net over F₃ — Digital

Digital (185, 204, 3590565)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁰⁴, 3590565, F₃, 2, 19) (dual of [(3590565, 2), 7180926, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²⁰⁴, 4194341, F₃, 2, 19) (dual of [(4194341, 2), 8388478, 20]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3²⁰³, 4194341, F₃, 2, 19) (dual of [(4194341, 2), 8388479, 20]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(3²², 40, F₃, 2, 9) (dual of [(40, 2), 58, 10]-NRT-code), using
        extracting embedded OOA [i] based on digital (13, 22, 40)-net over F₃, using
        trace code for nets [i] based on digital (2, 11, 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. linear OOA(3¹⁸¹, 4194301, F₃, 2, 19) (dual of [(4194301, 2), 8388421, 20]-NRT-code), using
        OOA 2-folding [i] based on linear OA(3¹⁸¹, 8388602, F₃, 19) (dual of [8388602, 8388421, 20]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁸¹, large, F₃, 19) (dual of [large, large−181, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(185, 185+19, large)-Net in Base 3 — Upper bound on s

There is no (185, 204, large)-net in base 3, because

17 times m-reduction [i] would yield (185, 187, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (185, 185+19, s)-Nets in Base 3

(185, 185+19, 932106)-Net over F3 — Constructive and digital

(185, 185+19, 3590565)-Net over F3 — Digital

(185, 185+19, large)-Net in Base 3 — Upper bound on s

(185, 185+19, 932106)-Net over F₃ — Constructive and digital

(185, 185+19, 3590565)-Net over F₃ — Digital