Best Known (219−160, 219, s)-Nets in Base 4
(219−160, 219, 66)-Net over F4 — Constructive and digital
Digital (59, 219, 66)-net over F4, using
- t-expansion [i] based on digital (49, 219, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- T6 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(219−160, 219, 91)-Net over F4 — Digital
Digital (59, 219, 91)-net over F4, using
- t-expansion [i] based on digital (50, 219, 91)-net over F4, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
(219−160, 219, 304)-Net over F4 — Upper bound on s (digital)
There is no digital (59, 219, 305)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4219, 305, F4, 160) (dual of [305, 86, 161]-code), but
- residual code [i] would yield OA(459, 144, S4, 40), but
- the linear programming bound shows that M ≥ 267 525938 139566 369814 585811 354664 265574 017946 230036 512794 828836 515572 789957 038790 441318 522730 123561 422710 374400 / 777 477803 949531 233536 373576 188373 355366 920576 587906 617821 062112 199463 182107 > 459 [i]
- residual code [i] would yield OA(459, 144, S4, 40), but
(219−160, 219, 390)-Net in Base 4 — Upper bound on s
There is no (59, 219, 391)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 724110 341184 134368 749416 435334 416425 832968 812673 031711 549979 282020 384542 143046 067417 399805 652875 825429 267614 704522 124825 599142 289112 > 4219 [i]