Best Known (78, 78+132, s)-Nets in Base 3
(78, 78+132, 53)-Net over F3 — Constructive and digital
Digital (78, 210, 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
(78, 78+132, 84)-Net over F3 — Digital
Digital (78, 210, 84)-net over F3, using
- t-expansion [i] based on digital (71, 210, 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
(78, 78+132, 347)-Net over F3 — Upper bound on s (digital)
There is no digital (78, 210, 348)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3210, 348, F3, 132) (dual of [348, 138, 133]-code), but
- residual code [i] would yield OA(378, 215, S3, 44), but
- 2 times truncation [i] would yield OA(376, 213, S3, 42), but
- the linear programming bound shows that M ≥ 24139 354386 854363 147378 476124 618112 504428 827791 667688 382538 767752 602863 057296 834401 801307 804700 / 12846 980782 406505 841743 696059 469546 157683 736221 588601 551817 > 376 [i]
- 2 times truncation [i] would yield OA(376, 213, S3, 42), but
- residual code [i] would yield OA(378, 215, S3, 44), but
(78, 78+132, 357)-Net in Base 3 — Upper bound on s
There is no (78, 210, 358)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 17270 912310 474454 217176 829130 003584 798563 154927 864414 479892 427279 607855 544951 116201 398441 482863 667757 > 3210 [i]