Best Known (78, 78+138, s)-Nets in Base 3
(78, 78+138, 53)-Net over F3 — Constructive and digital
Digital (78, 216, 53)-net over F3, using
- net from sequence [i] based on digital (78, 52)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 52)-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, 52)-sequence over F9, using
(78, 78+138, 84)-Net over F3 — Digital
Digital (78, 216, 84)-net over F3, using
- t-expansion [i] based on digital (71, 216, 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
(78, 78+138, 333)-Net over F3 — Upper bound on s (digital)
There is no digital (78, 216, 334)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3216, 334, F3, 138) (dual of [334, 118, 139]-code), but
- residual code [i] would yield OA(378, 195, S3, 46), but
- 4 times truncation [i] would yield OA(374, 191, S3, 42), but
- the linear programming bound shows that M ≥ 7425 373872 326041 246958 413277 162396 664697 868093 270451 580073 051722 485491 109207 030427 537458 984375 / 32911 891840 780579 755648 777011 603243 208306 897748 202179 983781 > 374 [i]
- 4 times truncation [i] would yield OA(374, 191, S3, 42), but
- residual code [i] would yield OA(378, 195, S3, 46), but
(78, 78+138, 349)-Net in Base 3 — Upper bound on s
There is no (78, 216, 350)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 13 271130 553323 791110 099066 933081 752406 403888 449132 029375 998785 582946 158431 488029 178409 021837 342083 823957 > 3216 [i]