Best Known (215−136, 215, s)-Nets in Base 3
(215−136, 215, 54)-Net over F3 — Constructive and digital
Digital (79, 215, 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
(215−136, 215, 84)-Net over F3 — Digital
Digital (79, 215, 84)-net over F3, using
- t-expansion [i] based on digital (71, 215, 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
(215−136, 215, 351)-Net over F3 — Upper bound on s (digital)
There is no digital (79, 215, 352)-net over F3, because
- 1 times m-reduction [i] would yield digital (79, 214, 352)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3214, 352, F3, 135) (dual of [352, 138, 136]-code), but
- residual code [i] would yield OA(379, 216, S3, 45), but
- 3 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]
- 3 times truncation [i] would yield OA(376, 213, S3, 42), but
- residual code [i] would yield OA(379, 216, S3, 45), but
- extracting embedded orthogonal array [i] would yield linear OA(3214, 352, F3, 135) (dual of [352, 138, 136]-code), but
(215−136, 215, 358)-Net in Base 3 — Upper bound on s
There is no (79, 215, 359)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4 190403 517685 166762 033606 380740 923085 260618 161101 457386 803075 748749 074492 971069 658215 368416 419683 016185 > 3215 [i]