Best Known (214−135, 214, s)-Nets in Base 3
(214−135, 214, 54)-Net over F3 — Constructive and digital
Digital (79, 214, 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
(214−135, 214, 84)-Net over F3 — Digital
Digital (79, 214, 84)-net over F3, using
- t-expansion [i] based on digital (71, 214, 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
(214−135, 214, 351)-Net over F3 — Upper bound on s (digital)
There is no digital (79, 214, 352)-net over F3, because
- 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
(214−135, 214, 361)-Net in Base 3 — Upper bound on s
There is no (79, 214, 362)-net in base 3, because
- 1 times m-reduction [i] would yield (79, 213, 362)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 472938 336709 390286 225551 889889 774411 515527 406033 296638 423170 339309 249484 198501 087734 953134 070120 645041 > 3213 [i]