Best Known (248−161, 248, s)-Nets in Base 3
(248−161, 248, 62)-Net over F3 — Constructive and digital
Digital (87, 248, 62)-net over F3, using
- net from sequence [i] based on digital (87, 61)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 61)-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, 61)-sequence over F9, using
(248−161, 248, 84)-Net over F3 — Digital
Digital (87, 248, 84)-net over F3, using
- t-expansion [i] based on digital (71, 248, 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
(248−161, 248, 303)-Net over F3 — Upper bound on s (digital)
There is no digital (87, 248, 304)-net over F3, because
- 2 times m-reduction [i] would yield digital (87, 246, 304)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3246, 304, F3, 159) (dual of [304, 58, 160]-code), but
- residual code [i] would yield OA(387, 144, S3, 53), but
- the linear programming bound shows that M ≥ 17223 084020 715556 855848 832250 872039 681441 493489 720994 373819 071391 134556 757302 805695 961783 787857 312694 961703 747535 992711 367021 / 52541 482507 385915 572483 476316 381490 546694 054744 799929 170031 348199 163698 205504 000000 > 387 [i]
- residual code [i] would yield OA(387, 144, S3, 53), but
- extracting embedded orthogonal array [i] would yield linear OA(3246, 304, F3, 159) (dual of [304, 58, 160]-code), but
(248−161, 248, 380)-Net in Base 3 — Upper bound on s
There is no (87, 248, 381)-net in base 3, because
- 1 times m-reduction [i] would yield (87, 247, 381)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 7675 164432 431909 440459 808853 962706 245452 104352 025674 631457 498424 788145 581830 839189 725409 370375 046788 031418 724675 144385 > 3247 [i]