MinT - Best Known (172, 172+32, s)-Nets in Base 3

Best Known (172, 172+32, s)-Nets in Base 3

(172, 172+32, 1480)-Net over F₃ — Constructive and digital

Digital (172, 204, 1480)-net over F₃, using

4 times m-reduction [i] based on digital (172, 208, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 52, 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]

(172, 172+32, 10164)-Net over F₃ — Digital

Digital (172, 204, 10164)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁴, 10164, F₃, 32) (dual of [10164, 9960, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3²⁰⁴, 19736, F₃, 32) (dual of [19736, 19532, 33]-code), using
  - constructionÂ X applied to C_e(31) âŠ‚ C_e(24) [i] based on
    1. 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]
    2. linear OA(3¹⁴⁵, 19683, F₃, 25) (dual of [19683, 19538, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    3. linear OA(3¹⁴, 53, F₃, 6) (dual of [53, 39, 7]-code), using
      - codes constructed by inverting constructionÂ Y₁ [i]

(172, 172+32, 4119107)-Net in Base 3 — Upper bound on s

There is no (172, 204, 4119108)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21 514801 732738 771496 736585 689358 148774 643905 018356 386956 661627 492810 698834 956651 613873 194534 294657 > 3²⁰⁴ [i]