Best Known (234−174, 234, s)-Nets in Base 4
(234−174, 234, 66)-Net over F4 — Constructive and digital
Digital (60, 234, 66)-net over F4, using
- t-expansion [i] based on digital (49, 234, 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
(234−174, 234, 91)-Net over F4 — Digital
Digital (60, 234, 91)-net over F4, using
- t-expansion [i] based on digital (50, 234, 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
(234−174, 234, 269)-Net over F4 — Upper bound on s (digital)
There is no digital (60, 234, 270)-net over F4, because
- 2 times m-reduction [i] would yield digital (60, 232, 270)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4232, 270, F4, 172) (dual of [270, 38, 173]-code), but
- residual code [i] would yield OA(460, 97, S4, 43), but
- the linear programming bound shows that M ≥ 397 009202 024702 523768 652504 484945 932069 120818 488153 315427 343594 355263 799296 / 269 558632 232160 195053 526197 508114 230285 > 460 [i]
- residual code [i] would yield OA(460, 97, S4, 43), but
- extracting embedded orthogonal array [i] would yield linear OA(4232, 270, F4, 172) (dual of [270, 38, 173]-code), but
(234−174, 234, 392)-Net in Base 4 — Upper bound on s
There is no (60, 234, 393)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 851 681891 605302 840398 371958 993223 295006 895607 820454 424353 618645 472750 944127 741071 987552 059721 989960 956650 746518 979727 789679 299875 649619 121600 > 4234 [i]