Best Known (80, 80+138, s)-Nets in Base 3
(80, 80+138, 55)-Net over F3 — Constructive and digital
Digital (80, 218, 55)-net over F3, using
- net from sequence [i] based on digital (80, 54)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 54)-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, 54)-sequence over F9, using
(80, 80+138, 84)-Net over F3 — Digital
Digital (80, 218, 84)-net over F3, using
- t-expansion [i] based on digital (71, 218, 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
(80, 80+138, 355)-Net over F3 — Upper bound on s (digital)
There is no digital (80, 218, 356)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3218, 356, F3, 138) (dual of [356, 138, 139]-code), but
- residual code [i] would yield OA(380, 217, S3, 46), but
- 4 times truncation [i] would yield OA(376, 213, S3, 42), but
- the linear programming bound shows that M ≥ 24139 354386 854363 147378 476124 618112 504428 827791 667688 382538 767752 602863 057296 834401 801307 804700 / 12846 980782 406505 841743 696059 469546 157683 736221 588601 551817 > 376 [i]
- 4 times truncation [i] would yield OA(376, 213, S3, 42), but
- residual code [i] would yield OA(380, 217, S3, 46), but
(80, 80+138, 362)-Net in Base 3 — Upper bound on s
There is no (80, 218, 363)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 114 197255 710049 673049 010031 933540 027860 039873 860510 033456 168610 409299 115005 570726 118262 127673 359697 277839 > 3218 [i]