MinT - Best Known (140−27, 140, s)-Nets in Base 3

Best Known (140−27, 140, s)-Nets in Base 3

Digital (113, 140, 688)-net over F₃, using

t-expansion [i] based on digital (112, 140, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 35, 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 (113, 140, 2264)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴⁰, 2264, F₃, 27) (dual of [2264, 2124, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴⁰, 2278, F₃, 27) (dual of [2278, 2138, 28]-code), using
  - 78 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0) [i] based on linear OA(3¹²⁶, 2186, F₃, 27) (dual of [2186, 2060, 28]-code), using
    - 1 times truncation [i] based on linear OA(3¹²⁷, 2187, F₃, 28) (dual of [2187, 2060, 29]-code), using
      - an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]

There is no (113, 140, 358005)-net in base 3, because

1 times m-reduction [i] would yield (113, 139, 358005)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 088646 522304 361863 472652 175950 635535 372147 571166 137278 892722 341339 > 3¹³⁹ [i]