MinT - Best Known (250−22, 250, s)-Nets in Base 3

Best Known (250−22, 250, s)-Nets in Base 3

(250−22, 250, 762744)-Net over F₃ — Constructive and digital

Digital (228, 250, 762744)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (28, 39, 144)-net over F₃, using
  - trace code for nets [i] based on digital (2, 13, 48)-net over F₂₇, using
    - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
        a function field by GarcÃa and Quoos [i]
2. digital (189, 211, 762600)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²¹¹, 762600, F₃, 22, 22) (dual of [(762600, 22), 16776989, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(3²¹¹, 8388600, F₃, 22) (dual of [8388600, 8388389, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹¹, large, F₃, 22) (dual of [large, large−211, 23]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(250−22, 250, 4194508)-Net over F₃ — Digital

Digital (228, 250, 4194508)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁵⁰, 4194508, F₃, 2, 22) (dual of [(4194508, 2), 8388766, 23]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3³⁹, 207, F₃, 2, 11) (dual of [(207, 2), 375, 12]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3³⁹, 207, F₃, 11) (dual of [207, 168, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(3³⁹, 256, F₃, 11) (dual of [256, 217, 12]-code), using
        constructionÂ X applied to C_e(10) âŠ‚ C_e(7) [i] based on
        linear OA(3³⁶, 243, F₃, 11) (dual of [243, 207, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(3²⁶, 243, F₃, 8) (dual of [243, 217, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]
  2. linear OOA(3²¹¹, 4194301, F₃, 2, 22) (dual of [(4194301, 2), 8388391, 23]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3²¹¹, 8388602, F₃, 22) (dual of [8388602, 8388391, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²¹¹, large, F₃, 22) (dual of [large, large−211, 23]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(250−22, 250, large)-Net in Base 3 — Upper bound on s

There is no (228, 250, large)-net in base 3, because

20 times m-reduction [i] would yield (228, 230, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (250−22, 250, s)-Nets in Base 3

(250−22, 250, 762744)-Net over F3 — Constructive and digital

(250−22, 250, 4194508)-Net over F3 — Digital

(250−22, 250, large)-Net in Base 3 — Upper bound on s

(250−22, 250, 762744)-Net over F₃ — Constructive and digital

(250−22, 250, 4194508)-Net over F₃ — Digital