MinT - Best Known (198−40, 198, s)-Nets in Base 3

Best Known (198−40, 198, s)-Nets in Base 3

(198−40, 198, 688)-Net over F₃ — Constructive and digital

Digital (158, 198, 688)-net over F₃, using

t-expansion [i] based on digital (157, 198, 688)-net over F₃, using
- 2 times m-reduction [i] based on digital (157, 200, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 50, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(198−40, 198, 2199)-Net over F₃ — Digital

Digital (158, 198, 2199)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹⁸, 2199, F₃, 40) (dual of [2199, 2001, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹⁸, 2231, F₃, 40) (dual of [2231, 2033, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(31) [i] based on
    1. linear OA(3¹⁸³, 2187, F₃, 40) (dual of [2187, 2004, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    2. linear OA(3¹⁴⁸, 2187, F₃, 32) (dual of [2187, 2039, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    3. linear OA(3¹⁵, 44, F₃, 7) (dual of [44, 29, 8]-code), using
      - generator matrices provided on Edelâ€™s homepage [i]

(198−40, 198, 219653)-Net in Base 3 — Upper bound on s

There is no (158, 198, 219654)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29514 039857 070980 832907 568356 692555 362036 562205 249846 758252 882518 212070 226009 804762 345701 484201 > 3¹⁹⁸ [i]