MinT - Best Known (165, 165+30, s)-Nets in Base 3

Best Known (165, 165+30, s)-Nets in Base 3

(165, 165+30, 1480)-Net over F₃ — Constructive and digital

Digital (165, 195, 1480)-net over F₃, using

t-expansion [i] based on digital (163, 195, 1480)-net over F₃, using
- 1 times m-reduction [i] based on digital (163, 196, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 49, 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]

(165, 165+30, 11395)-Net over F₃ — Digital

Digital (165, 195, 11395)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹⁵, 11395, F₃, 30) (dual of [11395, 11200, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹⁵, 19736, F₃, 30) (dual of [19736, 19541, 31]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(22) [i] based on
    1. linear OA(3¹⁸¹, 19683, F₃, 31) (dual of [19683, 19502, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(3¹³⁶, 19683, F₃, 23) (dual of [19683, 19547, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(3¹⁴, 53, F₃, 6) (dual of [53, 39, 7]-code), using
      - codes constructed by inverting constructionÂ Y₁ [i]

(165, 165+30, 5120492)-Net in Base 3 — Upper bound on s

There is no (165, 195, 5120493)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1093 063576 004059 465517 830601 856106 276447 642292 787483 090966 717878 041600 140748 821657 464235 299355 > 3¹⁹⁵ [i]