MinT - Best Known (250−47, 250, s)-Nets in Base 3

Best Known (250−47, 250, s)-Nets in Base 3

(250−47, 250, 896)-Net over F₃ — Constructive and digital

Digital (203, 250, 896)-net over F₃, using

3² times duplication [i] based on digital (201, 248, 896)-net over F₃, using
- t-expansion [i] based on digital (199, 248, 896)-net over F₃, using
  - trace code for nets [i] based on digital (13, 62, 224)-net over F₈₁, using
    - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
        a function field by SÃ©mirat [i]

(250−47, 250, 3805)-Net over F₃ — Digital

Digital (203, 250, 3805)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁵⁰, 3805, F₃, 47) (dual of [3805, 3555, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁵⁰, 6570, F₃, 47) (dual of [6570, 6320, 48]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3²⁴⁹, 6569, F₃, 47) (dual of [6569, 6320, 48]-code), using
    - constructionÂ X applied to C_e(46) âŠ‚ C_e(45) [i] based on
      1. linear OA(3²⁴⁹, 6561, F₃, 47) (dual of [6561, 6312, 48]-code), using an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]
      2. linear OA(3²⁴¹, 6561, F₃, 46) (dual of [6561, 6320, 47]-code), using an extension C_e(45) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
      3. linear OA(3⁰, 8, F₃, 0) (dual of [8, 8, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(250−47, 250, 689871)-Net in Base 3 — Upper bound on s

There is no (203, 250, 689872)-net in base 3, because

1 times m-reduction [i] would yield (203, 249, 689872)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 63561 943602 899053 188333 576385 023620 608626 441234 155967 641694 706603 170914 074146 205257 916937 395624 265054 362320 808096 633281 > 3²⁴⁹ [i]