MinT - Best Known (150, 150+41, s)-Nets in Base 3

Best Known (150, 150+41, s)-Nets in Base 3

(150, 150+41, 640)-Net over F₃ — Constructive and digital

Digital (150, 191, 640)-net over F₃, using

t-expansion [i] based on digital (149, 191, 640)-net over F₃, using
- 1 times m-reduction [i] based on digital (149, 192, 640)-net over F₃, using
  - trace code for nets [i] based on digital (5, 48, 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]

(150, 150+41, 1589)-Net over F₃ — Digital

Digital (150, 191, 1589)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹¹, 1589, F₃, 41) (dual of [1589, 1398, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹¹, 2195, F₃, 41) (dual of [2195, 2004, 42]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(3¹⁹⁰, 2194, F₃, 41) (dual of [2194, 2004, 42]-code), using
    - constructionÂ X applied to C_e(40) âŠ‚ C_e(39) [i] based on
      1. linear OA(3¹⁹⁰, 2187, F₃, 41) (dual of [2187, 1997, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
      2. linear OA(3¹⁸³, 2187, F₃, 40) (dual of [2187, 2004, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
      3. linear OA(3⁰, 7, F₃, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(150, 150+41, 141536)-Net in Base 3 — Upper bound on s

There is no (150, 191, 141537)-net in base 3, because

1 times m-reduction [i] would yield (150, 190, 141537)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 498539 620276 786901 697757 287484 408655 041002 141014 250513 401483 507594 426686 733381 357016 861329 > 3¹⁹⁰ [i]