MinT - Best Known (250−39, 250, s)-Nets in Base 3

Best Known (250−39, 250, s)-Nets in Base 3

(250−39, 250, 1500)-Net over F₃ — Constructive and digital

Digital (211, 250, 1500)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 30, 20)-net over F₃, using
  - net from sequence [i] based on digital (11, 19)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 9, N(F)Â =Â 19, and 1 place with degree 3 [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 (181, 220, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 55, 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]

(250−39, 250, 11871)-Net over F₃ — Digital

Digital (211, 250, 11871)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁵⁰, 11871, F₃, 39) (dual of [11871, 11621, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁵⁰, 19743, F₃, 39) (dual of [19743, 19493, 40]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(31) [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(3¹⁹⁰, 19683, F₃, 32) (dual of [19683, 19493, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    3. linear OA(3¹⁵, 60, F₃, 6) (dual of [60, 45, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁵, 85, F₃, 6) (dual of [85, 70, 7]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(250−39, 250, 7095515)-Net in Base 3 — Upper bound on s

There is no (211, 250, 7095516)-net in base 3, because

1 times m-reduction [i] would yield (211, 249, 7095516)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 63561 260443 066517 739802 404963 256517 360629 093310 598178 593031 370959 527188 941580 914251 250619 766315 572815 380656 051278 504465 > 3²⁴⁹ [i]