MinT - Best Known (199, 199+45, s)-Nets in Base 3

Best Known (199, 199+45, s)-Nets in Base 3

(199, 199+45, 1480)-Net over F₃ — Constructive and digital

Digital (199, 244, 1480)-net over F₃, using

trace code for nets [i] based on digital (16, 61, 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]

(199, 199+45, 4154)-Net over F₃ — Digital

Digital (199, 244, 4154)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴⁴, 4154, F₃, 45) (dual of [4154, 3910, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁴⁴, 6581, F₃, 45) (dual of [6581, 6337, 46]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3²⁴², 6579, F₃, 45) (dual of [6579, 6337, 46]-code), using
    - constructionÂ X applied to C([0,22]) âŠ‚ C([0,21]) [i] based on
      1. linear OA(3²⁴¹, 6562, F₃, 45) (dual of [6562, 6321, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,22], and minimum distance dÂ â‰¥ |{−22,−21,…,22}|+1Â =Â 46 (BCH-bound) [i]
      2. linear OA(3²²⁵, 6562, F₃, 43) (dual of [6562, 6337, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
      3. linear OA(3¹, 17, F₃, 1) (dual of [17, 16, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(199, 199+45, 843010)-Net in Base 3 — Upper bound on s

There is no (199, 244, 843011)-net in base 3, because

1 times m-reduction [i] would yield (199, 243, 843011)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 87 190324 902292 130460 519578 932937 441217 514931 048085 906044 840993 799885 967452 116469 321781 742205 821134 541428 977188 484861 > 3²⁴³ [i]