Best Known (159−103, 159, s)-Nets in Base 3
(159−103, 159, 48)-Net over F3 — Constructive and digital
Digital (56, 159, 48)-net over F3, using
- t-expansion [i] based on digital (45, 159, 48)-net over F3, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(159−103, 159, 64)-Net over F3 — Digital
Digital (56, 159, 64)-net over F3, using
- t-expansion [i] based on digital (49, 159, 64)-net over F3, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 49 and N(F) ≥ 64, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
(159−103, 159, 219)-Net over F3 — Upper bound on s (digital)
There is no digital (56, 159, 220)-net over F3, because
- 1 times m-reduction [i] would yield digital (56, 158, 220)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3158, 220, F3, 102) (dual of [220, 62, 103]-code), but
- residual code [i] would yield OA(356, 117, S3, 34), but
- the linear programming bound shows that M ≥ 6 572194 410340 770230 274451 771326 497100 923623 922000 075795 128289 692486 098341 171127 246711 375119 351943 651309 763751 930468 347431 822816 020354 752566 819295 644666 929761 907332 186743 642811 447530 849961 611050 200963 934251 142751 221668 590723 / 12308 202851 555111 921783 293103 347655 939532 914017 434792 921389 913805 577111 709332 831929 188280 965469 376469 390645 081450 650558 456420 708726 187952 770175 852049 285692 893702 393942 713983 331625 662904 794533 488932 > 356 [i]
- residual code [i] would yield OA(356, 117, S3, 34), but
- extracting embedded orthogonal array [i] would yield linear OA(3158, 220, F3, 102) (dual of [220, 62, 103]-code), but
(159−103, 159, 251)-Net in Base 3 — Upper bound on s
There is no (56, 159, 252)-net in base 3, because
- 1 times m-reduction [i] would yield (56, 158, 252)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2748 320859 871606 375913 964364 148868 185767 452370 036317 956990 918679 113397 920145 > 3158 [i]