MinT - Best Known (217−138, 217, s)-Nets in Base 3

Best Known (217−138, 217, s)-Nets in Base 3

Digital (79, 217, 54)-net over F₃, using

net from sequence [i] based on digital (79, 53)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 53)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

Digital (79, 217, 84)-net over F₃, using

There is no digital (79, 217, 344)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²¹⁷, 344, F₃, 138) (dual of [344, 127, 139]-code), but
- residual code [i] would yield OA(3⁷⁹, 205, S₃, 46), but
  - 4 times truncation [i] would yield OA(3⁷⁵, 201, S₃, 42), but
    - the linear programming bound shows that MÂ â‰¥ 70 125936 773107 976834 379244 843784 716640 335910 577502 309916 609029 015855 123273 912868 189563 572000 000000 / 109 901819 499502 271926 662008 620205 982068 132289 165070 372767 942049 > 3⁷⁵ [i]

There is no (79, 217, 356)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 36 142605 606337 882143 331074 964378 972297 756786 192103 058829 525439 609508 932343 605106 906287 716317 270873 370281 > 3²¹⁷ [i]