MinT - Best Known (161, 191, s)-Nets in Base 3

Best Known (161, 191, s)-Nets in Base 3

(161, 191, 1480)-Net over F₃ — Constructive and digital

Digital (161, 191, 1480)-net over F₃, using

t-expansion [i] based on digital (160, 191, 1480)-net over F₃, using
- 1 times m-reduction [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]

(161, 191, 9865)-Net over F₃ — Digital

Digital (161, 191, 9865)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁹¹, 9865, F₃, 2, 30) (dual of [(9865, 2), 19539, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁹¹, 19730, F₃, 30) (dual of [19730, 19539, 31]-code), using
  - 1 times truncation [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]

(161, 191, 3820137)-Net in Base 3 — Upper bound on s

There is no (161, 191, 3820138)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13 494625 212779 376688 172065 374147 055084 629217 291294 375668 051162 702484 560544 256586 726949 123961 > 3¹⁹¹ [i]