MinT - Best Known (227, 250, s)-Nets in Base 3

Best Known (227, 250, s)-Nets in Base 3

(227, 250, 762632)-Net over F₃ — Constructive and digital

Digital (227, 250, 762632)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 24, 32)-net over F₃, using
  - trace code for nets [i] based on digital (1, 12, 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 (203, 226, 762600)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²²⁶, 762600, F₃, 23, 23) (dual of [(762600, 23), 17539574, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(3²²⁶, 8388601, F₃, 23) (dual of [8388601, 8388375, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²²⁶, large, F₃, 23) (dual of [large, large−226, 24]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(227, 250, 3424344)-Net over F₃ — Digital

Digital (227, 250, 3424344)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁵⁰, 3424344, F₃, 2, 23) (dual of [(3424344, 2), 6848438, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²⁵⁰, 4194333, F₃, 2, 23) (dual of [(4194333, 2), 8388416, 24]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3²⁴, 32, F₃, 2, 11) (dual of [(32, 2), 40, 12]-NRT-code), using
      - extracting embedded OOA [i] based on digital (13, 24, 32)-net over F₃, using
        trace code for nets [i] based on digital (1, 12, 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(3²²⁶, 4194301, F₃, 2, 23) (dual of [(4194301, 2), 8388376, 24]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(3²²⁶, 8388602, F₃, 23) (dual of [8388602, 8388376, 24]-code), using
        discarding factors / shortening the dual code based on linear OA(3²²⁶, large, F₃, 23) (dual of [large, large−226, 24]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(227, 250, large)-Net in Base 3 — Upper bound on s

There is no (227, 250, large)-net in base 3, because

21 times m-reduction [i] would yield (227, 229, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]