MinT - Best Known (192, 238, s)-Nets in Base 3

Best Known (192, 238, s)-Nets in Base 3

(192, 238, 896)-Net over F₃ — Constructive and digital

Digital (192, 238, 896)-net over F₃, using

3² times duplication [i] based on digital (190, 236, 896)-net over F₃, using
- trace code for nets [i] based on digital (13, 59, 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]

(192, 238, 3164)-Net over F₃ — Digital

Digital (192, 238, 3164)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³⁸, 3164, F₃, 46) (dual of [3164, 2926, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3²³⁸, 3290, F₃, 46) (dual of [3290, 3052, 47]-code), using
  - constructionÂ X applied to C_e(45) âŠ‚ C_e(43) [i] based on
    1. linear OA(3²³⁷, 3281, F₃, 46) (dual of [3281, 3044, 47]-code), using an extension C_e(45) of the narrow-sense BCH-code C(I) with length 3280Â | 3⁸âˆ’1, defining interval IÂ =Â [1,45], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 46 [i]
    2. linear OA(3²²⁹, 3281, F₃, 44) (dual of [3281, 3052, 45]-code), using an extension C_e(43) of the narrow-sense BCH-code C(I) with length 3280Â | 3⁸âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
    3. linear OA(3¹, 9, F₃, 1) (dual of [9, 8, 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]

(192, 238, 407915)-Net in Base 3 — Upper bound on s

There is no (192, 238, 407916)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 358817 351315 352266 071872 984867 389363 341389 667829 808870 113513 184314 958057 462984 150889 213726 382915 313582 382909 734481 > 3²³⁸ [i]