Best Known (54, 54+100, s)-Nets in Base 3
(54, 54+100, 48)-Net over F3 — Constructive and digital
Digital (54, 154, 48)-net over F3, using
- t-expansion [i] based on digital (45, 154, 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
(54, 54+100, 64)-Net over F3 — Digital
Digital (54, 154, 64)-net over F3, using
- t-expansion [i] based on digital (49, 154, 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
(54, 54+100, 210)-Net over F3 — Upper bound on s (digital)
There is no digital (54, 154, 211)-net over F3, because
- 1 times m-reduction [i] would yield digital (54, 153, 211)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3153, 211, F3, 99) (dual of [211, 58, 100]-code), but
- residual code [i] would yield OA(354, 111, S3, 33), but
- the linear programming bound shows that M ≥ 69 521653 500476 614161 153568 439456 634621 343301 061108 091935 969781 672114 860568 911476 164860 756535 387360 911965 492756 797941 739348 801233 686914 510198 029737 162192 836257 832169 499658 440584 424171 453843 297237 778361 993190 002979 / 1 155137 652131 936103 891023 114978 691396 257144 646390 867107 634474 991714 089289 269885 604212 360556 654992 555516 006816 245080 116966 711030 040640 675877 351563 109531 582514 583401 847586 081113 972591 401217 > 354 [i]
- residual code [i] would yield OA(354, 111, S3, 33), but
- extracting embedded orthogonal array [i] would yield linear OA(3153, 211, F3, 99) (dual of [211, 58, 100]-code), but
(54, 54+100, 240)-Net in Base 3 — Upper bound on s
There is no (54, 154, 241)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 30 270603 763899 311789 524147 887200 526796 125236 690353 981331 237888 508605 958441 > 3154 [i]