MinT - Best Known (200−18, 200, s)-Nets in Base 3

Best Known (200−18, 200, s)-Nets in Base 3

(200−18, 200, 932099)-Net over F₃ — Constructive and digital

Digital (182, 200, 932099)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 20, 32)-net over F₃, using
  - trace code for nets [i] based on digital (1, 10, 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 (162, 180, 932067)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁸⁰, 932067, F₃, 18, 18) (dual of [(932067, 18), 16777026, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(3¹⁸⁰, large, F₃, 18) (dual of [large, large−180, 19]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(200−18, 200, 4194333)-Net over F₃ — Digital

Digital (182, 200, 4194333)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁰⁰, 4194333, F₃, 2, 18) (dual of [(4194333, 2), 8388466, 19]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3²⁰, 32, F₃, 2, 9) (dual of [(32, 2), 44, 10]-NRT-code), using
    - extracting embedded OOA [i] based on digital (11, 20, 32)-net over F₃, using
      - trace code for nets [i] based on digital (1, 10, 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. linear OOA(3¹⁸⁰, 4194301, F₃, 2, 18) (dual of [(4194301, 2), 8388422, 19]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹⁸⁰, 8388602, F₃, 18) (dual of [8388602, 8388422, 19]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁸⁰, large, F₃, 18) (dual of [large, large−180, 19]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(200−18, 200, large)-Net in Base 3 — Upper bound on s

There is no (182, 200, large)-net in base 3, because

16 times m-reduction [i] would yield (182, 184, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]