MinT - Best Known (245−40, 245, s)-Nets in Base 3

Best Known (245−40, 245, s)-Nets in Base 3

(245−40, 245, 1487)-Net over F₃ — Constructive and digital

Digital (205, 245, 1487)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 21, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (184, 224, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 56, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(245−40, 245, 9861)-Net over F₃ — Digital

Digital (205, 245, 9861)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁴⁵, 9861, F₃, 2, 40) (dual of [(9861, 2), 19477, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3²⁴⁵, 19722, F₃, 40) (dual of [19722, 19477, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(33) [i] based on
    1. linear OA(3²³⁵, 19683, F₃, 40) (dual of [19683, 19448, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    3. linear OA(3¹⁰, 39, F₃, 5) (dual of [39, 29, 6]-code), using
      - generator matrices provided on Edelâ€™s homepage [i]

(245−40, 245, 2904080)-Net in Base 3 — Upper bound on s

There is no (205, 245, 2904081)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 784 708086 323194 389415 388477 153879 342094 751894 475724 640183 249663 337321 223491 649832 309056 515362 122704 455642 734883 371537 > 3²⁴⁵ [i]