MinT - Best Known (157, 196, s)-Nets in Base 3

Best Known (157, 196, s)-Nets in Base 3

Digital (157, 196, 688)-net over F₃, using

4 times m-reduction [i] based on digital (157, 200, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 50, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

Digital (157, 196, 2296)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹⁶, 2296, F₃, 39) (dual of [2296, 2100, 40]-code), using
- 96 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 19 times 0, 1, 25 times 0) [i] based on linear OA(3¹⁸², 2186, F₃, 39) (dual of [2186, 2004, 40]-code), using
  - 1 times truncation [i] based on 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]

There is no (157, 196, 312558)-net in base 3, because

1 times m-reduction [i] would yield (157, 195, 312558)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1093 063618 541326 597155 305742 917754 567859 981404 181908 977283 417402 501654 139990 721413 219281 780129 > 3¹⁹⁵ [i]