MinT - Best Known (144, 144+39, s)-Nets in Base 3

Best Known (144, 144+39, s)-Nets in Base 3

(144, 144+39, 640)-Net over F₃ — Constructive and digital

Digital (144, 183, 640)-net over F₃, using

t-expansion [i] based on digital (143, 183, 640)-net over F₃, using
- 1 times m-reduction [i] based on digital (143, 184, 640)-net over F₃, using
  - trace code for nets [i] based on digital (5, 46, 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]

(144, 144+39, 1595)-Net over F₃ — Digital

Digital (144, 183, 1595)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸³, 1595, F₃, 39) (dual of [1595, 1412, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸³, 2188, F₃, 39) (dual of [2188, 2005, 40]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,19], and minimum distance dÂ â‰¥ |{−19,−18,…,19}|+1Â =Â 40 (BCH-bound) [i]

(144, 144+39, 147384)-Net in Base 3 — Upper bound on s

There is no (144, 183, 147385)-net in base 3, because

1 times m-reduction [i] would yield (144, 182, 147385)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 685 682964 934696 110400 287127 057500 768976 651798 078615 287406 798207 815608 384879 050467 737211 > 3¹⁸² [i]