Best Known (51, 51+90, s)-Nets in Base 3
(51, 51+90, 48)-Net over F3 — Constructive and digital
Digital (51, 141, 48)-net over F3, using
- t-expansion [i] based on digital (45, 141, 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
(51, 51+90, 64)-Net over F3 — Digital
Digital (51, 141, 64)-net over F3, using
- t-expansion [i] based on digital (49, 141, 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
(51, 51+90, 213)-Net over F3 — Upper bound on s (digital)
There is no digital (51, 141, 214)-net over F3, because
- extracting embedded orthogonal array [i] would yield linear OA(3141, 214, F3, 90) (dual of [214, 73, 91]-code), but
- residual code [i] would yield OA(351, 123, S3, 30), but
- the linear programming bound shows that M ≥ 184 562452 634065 452157 662353 211684 517149 197249 714498 133253 737898 233581 110360 924816 302833 420114 709971 475375 / 80 717016 244069 464587 443624 242700 434296 750214 169562 345287 177003 948378 401931 804237 > 351 [i]
- residual code [i] would yield OA(351, 123, S3, 30), but
(51, 51+90, 233)-Net in Base 3 — Upper bound on s
There is no (51, 141, 234)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 19 556780 310906 930474 194800 551358 210749 903478 546922 548403 675025 323037 > 3141 [i]