MinT - Best Known (218−20, 218, s)-Nets in Base 3

Best Known (218−20, 218, s)-Nets in Base 3

(218−20, 218, 838892)-Net over F₃ — Constructive and digital

Digital (198, 218, 838892)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (12, 22, 32)-net over F₃, using
  - trace code for nets [i] based on digital (1, 11, 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 (176, 196, 838860)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁹⁶, 838860, F₃, 20, 20) (dual of [(838860, 20), 16777004, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(3¹⁹⁶, 8388600, F₃, 20) (dual of [8388600, 8388404, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁹⁶, large, F₃, 20) (dual of [large, large−196, 21]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(218−20, 218, 4141523)-Net over F₃ — Digital

Digital (198, 218, 4141523)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²¹⁸, 4141523, F₃, 2, 20) (dual of [(4141523, 2), 8282828, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²¹⁸, 4194333, F₃, 2, 20) (dual of [(4194333, 2), 8388448, 21]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3²², 32, F₃, 2, 10) (dual of [(32, 2), 42, 11]-NRT-code), using
      - extracting embedded OOA [i] based on digital (12, 22, 32)-net over F₃, using
        trace code for nets [i] based on digital (1, 11, 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(3¹⁹⁶, 4194301, F₃, 2, 20) (dual of [(4194301, 2), 8388406, 21]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(3¹⁹⁶, 8388602, F₃, 20) (dual of [8388602, 8388406, 21]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁹⁶, large, F₃, 20) (dual of [large, large−196, 21]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(218−20, 218, large)-Net in Base 3 — Upper bound on s

There is no (198, 218, large)-net in base 3, because

18 times m-reduction [i] would yield (198, 200, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]