MinT - Best Known (250−41, 250, s)-Nets in Base 3

Best Known (250−41, 250, s)-Nets in Base 3

(250−41, 250, 1488)-Net over F₃ — Constructive and digital

Digital (209, 250, 1488)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 22, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (187, 228, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 57, 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]

(250−41, 250, 9727)-Net over F₃ — Digital

Digital (209, 250, 9727)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁵⁰, 9727, F₃, 2, 41) (dual of [(9727, 2), 19204, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²⁵⁰, 9858, F₃, 2, 41) (dual of [(9858, 2), 19466, 42]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²⁵⁰, 19716, F₃, 41) (dual of [19716, 19466, 42]-code), using
    - constructionÂ X applied to C_e(40) âŠ‚ C_e(36) [i] based on
      1. linear OA(3²⁴⁴, 19683, F₃, 41) (dual of [19683, 19439, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
      2. 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]
      3. linear OA(3⁶, 33, F₃, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
        discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(250−41, 250, 3617708)-Net in Base 3 — Upper bound on s

There is no (209, 250, 3617709)-net in base 3, because

1 times m-reduction [i] would yield (209, 249, 3617709)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 63561 585299 954412 433075 766214 868333 997913 626373 451261 904662 584098 954365 936117 142146 275026 010156 482395 311753 392735 152977 > 3²⁴⁹ [i]