MinT - Best Known (192, 192+44, s)-Nets in Base 3

Best Known (192, 192+44, s)-Nets in Base 3

(192, 192+44, 896)-Net over F₃ — Constructive and digital

Digital (192, 236, 896)-net over F₃, using

t-expansion [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, 192+44, 3818)-Net over F₃ — Digital

Digital (192, 236, 3818)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³⁶, 3818, F₃, 44) (dual of [3818, 3582, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3²³⁶, 6574, F₃, 44) (dual of [6574, 6338, 45]-code), using
  - constructionÂ X applied to C_e(43) âŠ‚ C_e(40) [i] based on
    1. linear OA(3²³³, 6561, F₃, 44) (dual of [6561, 6328, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
    2. linear OA(3²¹⁷, 6561, F₃, 41) (dual of [6561, 6344, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(192, 192+44, 594315)-Net in Base 3 — Upper bound on s

There is no (192, 236, 594316)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 39867 796784 922407 253848 552682 750146 458985 134976 195260 276177 488188 139615 671481 136844 883949 677980 606260 766151 283977 > 3²³⁶ [i]