MinT - Best Known (200, 236, s)-Nets in Base 3

Best Known (200, 236, s)-Nets in Base 3

(200, 236, 1499)-Net over F₃ — Constructive and digital

Digital (200, 236, 1499)-net over F₃, using

3¹ times duplication [i] based on digital (199, 235, 1499)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 27, 19)-net over F₃, using
    - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
        an explicitly constructive algebraic function field [i]
  2. digital (172, 208, 1480)-net over F₃, using
    - trace code for nets [i] based on digital (16, 52, 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]

(200, 236, 13401)-Net over F₃ — Digital

Digital (200, 236, 13401)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³⁶, 13401, F₃, 36) (dual of [13401, 13165, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3²³⁶, 19756, F₃, 36) (dual of [19756, 19520, 37]-code), using
  - 3 times code embedding in larger space [i] based on linear OA(3²³³, 19753, F₃, 36) (dual of [19753, 19520, 37]-code), using
    - constructionÂ X applied to C_e(36) âŠ‚ C_e(27) [i] based on
      1. linear OA(3²¹⁷, 19683, F₃, 37) (dual of [19683, 19466, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
      2. linear OA(3¹⁶³, 19683, F₃, 28) (dual of [19683, 19520, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      3. linear OA(3¹⁶, 70, F₃, 7) (dual of [70, 54, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁶, 82, F₃, 7) (dual of [82, 66, 8]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(200, 236, 6802816)-Net in Base 3 — Upper bound on s

There is no (200, 236, 6802817)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 39867 334611 365206 040527 879599 133041 453983 820247 135818 459653 966177 713133 168268 958342 743701 065480 029081 314789 230665 > 3²³⁶ [i]