MinT - Best Known (136, 169, s)-Nets in Base 3

Best Known (136, 169, s)-Nets in Base 3

Digital (136, 169, 688)-net over F₃, using

3 times m-reduction [i] based on digital (136, 172, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 43, 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 (136, 169, 2302)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶⁹, 2302, F₃, 33) (dual of [2302, 2133, 34]-code), using
- 101 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 8 times 0, 1, 11 times 0, 1, 14 times 0, 1, 19 times 0, 1, 25 times 0) [i] based on linear OA(3¹⁵⁴, 2186, F₃, 33) (dual of [2186, 2032, 34]-code), using
  - 1 times truncation [i] based on linear OA(3¹⁵⁵, 2187, F₃, 34) (dual of [2187, 2032, 35]-code), using
    - an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

There is no (136, 169, 347747)-net in base 3, because

1 times m-reduction [i] would yield (136, 168, 347747)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 143 345584 278029 104535 994502 614899 109094 564495 219005 746434 167726 591071 701626 224865 > 3¹⁶⁸ [i]