Best Known (218−139, 218, s)-Nets in Base 3
(218−139, 218, 54)-Net over F3 — Constructive and digital
Digital (79, 218, 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
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 13 and N(F) ≥ 64, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- base reduction for sequences [i] based on digital (13, 53)-sequence over F9, using
(218−139, 218, 84)-Net over F3 — Digital
Digital (79, 218, 84)-net over F3, using
- t-expansion [i] based on digital (71, 218, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(218−139, 218, 343)-Net over F3 — Upper bound on s (digital)
There is no digital (79, 218, 344)-net over F3, because
- 1 times m-reduction [i] would yield digital (79, 217, 344)-net over F3, but
- 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 M ≥ 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]
- 4 times truncation [i] would yield OA(375, 201, S3, 42), but
- residual code [i] would yield OA(379, 205, S3, 46), but
- extracting embedded orthogonal array [i] would yield linear OA(3217, 344, F3, 138) (dual of [344, 127, 139]-code), but
(218−139, 218, 355)-Net in Base 3 — Upper bound on s
There is no (79, 218, 356)-net in base 3, because
- 1 times m-reduction [i] would yield (79, 217, 356)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 36 142605 606337 882143 331074 964378 972297 756786 192103 058829 525439 609508 932343 605106 906287 716317 270873 370281 > 3217 [i]