MinT - Best Known (154, 154+16, s)-Nets in Base 3

Best Known (154, 154+16, s)-Nets in Base 3

(154, 154+16, 1048607)-Net over F₃ — Constructive and digital

Digital (154, 170, 1048607)-net over F₃, using

3¹ times duplication [i] based on digital (153, 169, 1048607)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 18, 32)-net over F₃, using
    - trace code for nets [i] based on digital (1, 9, 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 (135, 151, 1048575)-net over F₃, using
    - net defined by OOA [i] based on linear OOA(3¹⁵¹, 1048575, F₃, 16, 16) (dual of [(1048575, 16), 16777049, 17]-NRT-code), using
      - OA 8-folding and stacking [i] based on linear OA(3¹⁵¹, 8388600, F₃, 16) (dual of [8388600, 8388449, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁵¹, large, F₃, 16) (dual of [large, large−151, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(154, 154+16, 4151926)-Net over F₃ — Digital

Digital (154, 170, 4151926)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷⁰, 4151926, F₃, 2, 16) (dual of [(4151926, 2), 8303682, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹⁷⁰, 4194333, F₃, 2, 16) (dual of [(4194333, 2), 8388496, 17]-NRT-code), using
  - 3¹ times duplication [i] based on linear OOA(3¹⁶⁹, 4194333, F₃, 2, 16) (dual of [(4194333, 2), 8388497, 17]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(3¹⁸, 32, F₃, 2, 8) (dual of [(32, 2), 46, 9]-NRT-code), using
        extracting embedded OOA [i] based on digital (10, 18, 32)-net over F₃, using
        trace code for nets [i] based on digital (1, 9, 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(3¹⁵¹, 4194301, F₃, 2, 16) (dual of [(4194301, 2), 8388451, 17]-NRT-code), using
        OOA 2-folding [i] based on linear OA(3¹⁵¹, 8388602, F₃, 16) (dual of [8388602, 8388451, 17]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁵¹, large, F₃, 16) (dual of [large, large−151, 17]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(154, 154+16, large)-Net in Base 3 — Upper bound on s

There is no (154, 170, large)-net in base 3, because

14 times m-reduction [i] would yield (154, 156, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (154, 154+16, s)-Nets in Base 3

(154, 154+16, 1048607)-Net over F3 — Constructive and digital

(154, 154+16, 4151926)-Net over F3 — Digital

(154, 154+16, large)-Net in Base 3 — Upper bound on s

(154, 154+16, 1048607)-Net over F₃ — Constructive and digital

(154, 154+16, 4151926)-Net over F₃ — Digital