MinT - Best Known (206, 206+40, s)-Nets in Base 3

Best Known (206, 206+40, s)-Nets in Base 3

(206, 206+40, 1488)-Net over F₃ — Constructive and digital

Digital (206, 246, 1488)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 22, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (184, 224, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 56, 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]

(206, 206+40, 9865)-Net over F₃ — Digital

Digital (206, 246, 9865)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁴⁶, 9865, F₃, 2, 40) (dual of [(9865, 2), 19484, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3²⁴⁶, 19730, F₃, 40) (dual of [19730, 19484, 41]-code), using
  - constructionÂ X applied to C_e(39) âŠ‚ C_e(33) [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(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    3. linear OA(3¹¹, 47, F₃, 5) (dual of [47, 36, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(206, 206+40, 3068067)-Net in Base 3 — Upper bound on s

There is no (206, 246, 3068068)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2354 128264 455914 038312 337659 969664 707211 097780 036707 878923 614810 903612 478861 473137 001304 673082 200840 944913 351781 029953 > 3²⁴⁶ [i]