MinT - Best Known (198, 198+38, s)-Nets in Base 3

Best Known (198, 198+38, s)-Nets in Base 3

(198, 198+38, 1487)-Net over F₃ — Constructive and digital

Digital (198, 236, 1487)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (178, 216, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 54, 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]

(198, 198+38, 9861)-Net over F₃ — Digital

Digital (198, 236, 9861)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²³⁶, 9861, F₃, 2, 38) (dual of [(9861, 2), 19486, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3²³⁶, 19722, F₃, 38) (dual of [19722, 19486, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(31) [i] based on
    1. linear OA(3²²⁶, 19683, F₃, 38) (dual of [19683, 19457, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(3¹⁹⁰, 19683, F₃, 32) (dual of [19683, 19493, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    3. linear OA(3¹⁰, 39, F₃, 5) (dual of [39, 29, 6]-code), using
      - generator matrices provided on Edelâ€™s homepage [i]

(198, 198+38, 3346041)-Net in Base 3 — Upper bound on s

There is no (198, 236, 3346042)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 39867 342790 457752 716230 459844 539080 216154 012003 497363 454985 553952 642010 353145 122475 254668 724400 204812 213748 612945 > 3²³⁶ [i]