Best Known (81, 81+143, s)-Nets in Base 3
(81, 81+143, 56)-Net over F3 — Constructive and digital
Digital (81, 224, 56)-net over F3, using
- net from sequence [i] based on digital (81, 55)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 55)-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, 55)-sequence over F9, using
(81, 81+143, 84)-Net over F3 — Digital
Digital (81, 224, 84)-net over F3, using
- t-expansion [i] based on digital (71, 224, 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
(81, 81+143, 359)-Net over F3 — Upper bound on s (digital)
There is no digital (81, 224, 360)-net over F3, because
- 2 times m-reduction [i] would yield digital (81, 222, 360)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3222, 360, F3, 141) (dual of [360, 138, 142]-code), but
- residual code [i] would yield OA(381, 218, S3, 47), but
- 5 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]
- 5 times truncation [i] would yield OA(376, 213, S3, 42), but
- residual code [i] would yield OA(381, 218, S3, 47), but
- extracting embedded orthogonal array [i] would yield linear OA(3222, 360, F3, 141) (dual of [360, 138, 142]-code), but
(81, 81+143, 363)-Net in Base 3 — Upper bound on s
There is no (81, 224, 364)-net in base 3, because
- 1 times m-reduction [i] would yield (81, 223, 364)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 26626 349459 704352 670038 781304 546915 218298 853711 277596 983555 064268 389182 671873 630777 112196 530085 194636 275921 > 3223 [i]