MinT - Best Known (192−30, 192, s)-Nets in Base 3

Best Known (192−30, 192, s)-Nets in Base 3

(192−30, 192, 1480)-Net over F₃ — Constructive and digital

Digital (162, 192, 1480)-net over F₃, using

t-expansion [i] based on digital (160, 192, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 48, 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]

(192−30, 192, 10127)-Net over F₃ — Digital

Digital (162, 192, 10127)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹², 10127, F₃, 30) (dual of [10127, 9935, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹², 19731, F₃, 30) (dual of [19731, 19539, 31]-code), using
  - strength reduction [i] based on linear OA(3¹⁹², 19731, F₃, 31) (dual of [19731, 19539, 32]-code), using
    - constructionÂ X applied to C([0,15]) âŠ‚ C([0,12]) [i] based on
      1. linear OA(3¹⁸¹, 19684, F₃, 31) (dual of [19684, 19503, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
      2. linear OA(3¹⁴⁵, 19684, F₃, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684Â | 3¹⁸âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [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]

(192−30, 192, 4110429)-Net in Base 3 — Upper bound on s

There is no (162, 192, 4110430)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40 483882 306344 471727 938020 748394 270018 188331 185032 057861 753966 320642 076558 558880 825606 817257 > 3¹⁹² [i]