MinT - Best Known (198, 244, s)-Nets in Base 3

Best Known (198, 244, s)-Nets in Base 3

(198, 244, 896)-Net over F₃ — Constructive and digital

Digital (198, 244, 896)-net over F₃, using

t-expansion [i] based on digital (196, 244, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 61, 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]

(198, 244, 3681)-Net over F₃ — Digital

Digital (198, 244, 3681)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴⁴, 3681, F₃, 46) (dual of [3681, 3437, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁴⁴, 6574, F₃, 46) (dual of [6574, 6330, 47]-code), using
  - constructionÂ X applied to C_e(45) âŠ‚ C_e(42) [i] based on
    1. 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]
    2. linear OA(3²²⁵, 6561, F₃, 43) (dual of [6561, 6336, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(198, 244, 543303)-Net in Base 3 — Upper bound on s

There is no (198, 244, 543304)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 261 574787 351912 497262 653891 004061 905062 665983 519143 560679 947604 486802 842980 087159 609557 291123 609787 190561 966908 789345 > 3²⁴⁴ [i]