Best Known (59, 59+105, s)-Nets in Base 3
(59, 59+105, 48)-Net over F3 — Constructive and digital
Digital (59, 164, 48)-net over F3, using
- t-expansion [i] based on digital (45, 164, 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
(59, 59+105, 64)-Net over F3 — Digital
Digital (59, 164, 64)-net over F3, using
- t-expansion [i] based on digital (49, 164, 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
(59, 59+105, 241)-Net over F3 — Upper bound on s (digital)
There is no digital (59, 164, 242)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3164, 242, F3, 105) (dual of [242, 78, 106]-code), but
- residual code [i] would yield OA(359, 136, S3, 35), but
- the linear programming bound shows that M ≥ 22 726839 204691 891253 644696 898369 207187 081234 181735 714843 341275 897616 762362 710018 187521 018627 172327 868695 031408 531569 772970 640083 192156 200366 308595 435159 925930 757650 433436 609239 366400 / 1549 254697 196676 010642 793897 637084 831954 187799 248006 962271 072379 219020 389873 109263 689496 974500 056880 902187 125508 048087 135151 525741 030050 717474 931926 817179 > 359 [i]
- residual code [i] would yield OA(359, 136, S3, 35), but
(59, 59+105, 268)-Net in Base 3 — Upper bound on s
There is no (59, 164, 269)-net in base 3, because
- 1 times m-reduction [i] would yield (59, 163, 269)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 665315 953937 050030 563908 944821 068158 967384 630467 761506 863657 396693 304768 944593 > 3163 [i]