MinT - Best Known (141, 141+38, s)-Nets in Base 3

Best Known (141, 141+38, s)-Nets in Base 3

(141, 141+38, 640)-Net over F₃ — Constructive and digital

Digital (141, 179, 640)-net over F₃, using

t-expansion [i] based on digital (140, 179, 640)-net over F₃, using
- 1 times m-reduction [i] based on digital (140, 180, 640)-net over F₃, using
  - trace code for nets [i] based on digital (5, 45, 160)-net over F₈₁, using
    - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
        a function field by SÃ©mirat [i]

(141, 141+38, 1599)-Net over F₃ — Digital

Digital (141, 179, 1599)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁷⁹, 1599, F₃, 38) (dual of [1599, 1420, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁷⁹, 2200, F₃, 38) (dual of [2200, 2021, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(34) [i] based on
    1. linear OA(3¹⁷⁶, 2187, F₃, 38) (dual of [2187, 2011, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(3¹⁶², 2187, F₃, 35) (dual of [2187, 2025, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(141, 141+38, 123909)-Net in Base 3 — Upper bound on s

There is no (141, 179, 123910)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25 393811 355949 202749 348310 053100 837104 437963 665372 649336 195680 172607 221586 833890 607681 > 3¹⁷⁹ [i]