MinT - Best Known (238−40, 238, s)-Nets in Base 3

Best Known (238−40, 238, s)-Nets in Base 3

(238−40, 238, 1480)-Net over F₃ — Constructive and digital

Digital (198, 238, 1480)-net over F₃, using

t-expansion [i] based on digital (196, 238, 1480)-net over F₃, using
- 2 times m-reduction [i] based on digital (196, 240, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 60, 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]

(238−40, 238, 8058)-Net over F₃ — Digital

Digital (198, 238, 8058)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²³⁸, 8058, F₃, 2, 40) (dual of [(8058, 2), 15878, 41]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²³⁸, 9848, F₃, 2, 40) (dual of [(9848, 2), 19458, 41]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²³⁸, 19696, F₃, 40) (dual of [19696, 19458, 41]-code), using
    - constructionÂ X applied to C_e(39) âŠ‚ C_e(36) [i] based on
      1. linear OA(3²³⁵, 19683, F₃, 40) (dual of [19683, 19448, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
      2. linear OA(3²¹⁷, 19683, F₃, 37) (dual of [19683, 19466, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
      3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]

(238−40, 238, 1977037)-Net in Base 3 — Upper bound on s

There is no (198, 238, 1977038)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 358807 327177 019534 233771 218130 071868 518823 290679 333706 837226 343695 382948 073673 576073 858016 217543 910482 828455 732649 > 3²³⁸ [i]