Best Known (199−143, 199, s)-Nets in Base 4
(199−143, 199, 66)-Net over F4 — Constructive and digital
Digital (56, 199, 66)-net over F4, using
- t-expansion [i] based on digital (49, 199, 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
(199−143, 199, 91)-Net over F4 — Digital
Digital (56, 199, 91)-net over F4, using
- t-expansion [i] based on digital (50, 199, 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
(199−143, 199, 344)-Net over F4 — Upper bound on s (digital)
There is no digital (56, 199, 345)-net over F4, because
- 3 times m-reduction [i] would yield digital (56, 196, 345)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4196, 345, F4, 140) (dual of [345, 149, 141]-code), but
- residual code [i] would yield OA(456, 204, S4, 35), but
- the linear programming bound shows that M ≥ 78295 866271 954113 495370 355913 581218 786159 455204 222712 111589 032673 922357 657600 / 14 695871 573792 476851 963908 736113 155530 258961 > 456 [i]
- residual code [i] would yield OA(456, 204, S4, 35), but
- extracting embedded orthogonal array [i] would yield linear OA(4196, 345, F4, 140) (dual of [345, 149, 141]-code), but
(199−143, 199, 378)-Net in Base 4 — Upper bound on s
There is no (56, 199, 379)-net in base 4, because
- 1 times m-reduction [i] would yield (56, 198, 379)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 185142 893133 861575 750058 796333 311597 129050 342782 744674 762284 038076 822274 561843 564274 421112 341181 827111 721242 225209 877248 > 4198 [i]