Best Known (63, 175, s)-Nets in Base 3
(63, 175, 48)-Net over F3 — Constructive and digital
Digital (63, 175, 48)-net over F3, using
- t-expansion [i] based on digital (45, 175, 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
(63, 175, 64)-Net over F3 — Digital
Digital (63, 175, 64)-net over F3, using
- t-expansion [i] based on digital (49, 175, 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
(63, 175, 260)-Net over F3 — Upper bound on s (digital)
There is no digital (63, 175, 261)-net over F3, because
- 1 times m-reduction [i] would yield digital (63, 174, 261)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3174, 261, F3, 111) (dual of [261, 87, 112]-code), but
- residual code [i] would yield OA(363, 149, S3, 37), but
- the linear programming bound shows that M ≥ 46 631470 124673 049327 400235 264470 858493 236508 054985 173899 541299 524620 475842 712041 549727 037290 684522 127992 940560 161444 675369 235323 506654 361935 251741 549444 156200 159989 526202 879027 434375 / 37 524831 135040 905996 809144 831977 213264 438373 079442 262970 729827 862827 594743 088766 089529 538795 972448 769675 682777 166896 727335 845836 387566 113951 189899 042231 > 363 [i]
- residual code [i] would yield OA(363, 149, S3, 37), but
- extracting embedded orthogonal array [i] would yield linear OA(3174, 261, F3, 111) (dual of [261, 87, 112]-code), but
(63, 175, 284)-Net in Base 3 — Upper bound on s
There is no (63, 175, 285)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 357396 589382 199705 882503 713957 739649 542089 267746 965412 996495 595697 234574 628997 669025 > 3175 [i]