Best Known (213−135, 213, s)-Nets in Base 3
(213−135, 213, 53)-Net over F3 — Constructive and digital
Digital (78, 213, 53)-net over F3, using
- net from sequence [i] based on digital (78, 52)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 52)-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, 52)-sequence over F9, using
(213−135, 213, 84)-Net over F3 — Digital
Digital (78, 213, 84)-net over F3, using
- t-expansion [i] based on digital (71, 213, 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
(213−135, 213, 339)-Net over F3 — Upper bound on s (digital)
There is no digital (78, 213, 340)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3213, 340, F3, 135) (dual of [340, 127, 136]-code), but
- residual code [i] would yield OA(378, 204, S3, 45), but
- 3 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]
- 3 times truncation [i] would yield OA(375, 201, S3, 42), but
- residual code [i] would yield OA(378, 204, S3, 45), but
(213−135, 213, 354)-Net in Base 3 — Upper bound on s
There is no (78, 213, 355)-net in base 3, because
- 1 times m-reduction [i] would yield (78, 212, 355)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 153682 638497 091372 765762 020194 669987 790428 343146 942479 785446 539364 477072 786318 404044 018327 076824 008107 > 3212 [i]