Best Known (210−151, 210, s)-Nets in Base 4
(210−151, 210, 66)-Net over F4 — Constructive and digital
Digital (59, 210, 66)-net over F4, using
- t-expansion [i] based on digital (49, 210, 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
(210−151, 210, 91)-Net over F4 — Digital
Digital (59, 210, 91)-net over F4, using
- t-expansion [i] based on digital (50, 210, 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
(210−151, 210, 357)-Net over F4 — Upper bound on s (digital)
There is no digital (59, 210, 358)-net over F4, because
- 3 times m-reduction [i] would yield digital (59, 207, 358)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4207, 358, F4, 148) (dual of [358, 151, 149]-code), but
- residual code [i] would yield OA(459, 209, S4, 37), but
- the linear programming bound shows that M ≥ 3446 699662 509398 967731 365644 377802 984275 634503 925011 721737 994240 000000 000000 / 10213 023598 987442 955247 667333 699389 861739 > 459 [i]
- residual code [i] would yield OA(459, 209, S4, 37), but
- extracting embedded orthogonal array [i] would yield linear OA(4207, 358, F4, 148) (dual of [358, 151, 149]-code), but
(210−151, 210, 397)-Net in Base 4 — Upper bound on s
There is no (59, 210, 398)-net in base 4, because
- 1 times m-reduction [i] would yield (59, 209, 398)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 764264 725725 044001 933691 725014 397788 263408 876229 112080 527024 584784 309917 631821 513713 318617 185272 571253 169799 610755 743202 599044 > 4209 [i]