Best Known (59, 59+104, s)-Nets in Base 3
(59, 59+104, 48)-Net over F3 — Constructive and digital
Digital (59, 163, 48)-net over F3, using
- t-expansion [i] based on digital (45, 163, 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+104, 64)-Net over F3 — Digital
Digital (59, 163, 64)-net over F3, using
- t-expansion [i] based on digital (49, 163, 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+104, 251)-Net over F3 — Upper bound on s (digital)
There is no digital (59, 163, 252)-net over F3, because
- 2 times m-reduction [i] would yield digital (59, 161, 252)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3161, 252, F3, 102) (dual of [252, 91, 103]-code), but
- residual code [i] would yield OA(359, 149, S3, 34), but
- the linear programming bound shows that M ≥ 161072 631961 680185 280440 930117 244701 258542 024979 482340 755321 093357 664942 075114 897456 280621 705350 294957 376375 / 10 335291 754987 667559 057108 370888 319896 679664 836625 228437 457068 286900 284102 491501 > 359 [i]
- residual code [i] would yield OA(359, 149, S3, 34), but
- extracting embedded orthogonal array [i] would yield linear OA(3161, 252, F3, 102) (dual of [252, 91, 103]-code), but
(59, 59+104, 268)-Net in Base 3 — Upper bound on s
There is no (59, 163, 269)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 665315 953937 050030 563908 944821 068158 967384 630467 761506 863657 396693 304768 944593 > 3163 [i]