Best Known (60, 60+173, s)-Nets in Base 4
(60, 60+173, 66)-Net over F4 — Constructive and digital
Digital (60, 233, 66)-net over F4, using
- t-expansion [i] based on digital (49, 233, 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
(60, 60+173, 91)-Net over F4 — Digital
Digital (60, 233, 91)-net over F4, using
- t-expansion [i] based on digital (50, 233, 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
(60, 60+173, 269)-Net over F4 — Upper bound on s (digital)
There is no digital (60, 233, 270)-net over F4, because
- 1 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
(60, 60+173, 393)-Net in Base 4 — Upper bound on s
There is no (60, 233, 394)-net in base 4, because
- 1 times m-reduction [i] would yield (60, 232, 394)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 56 198617 200464 028656 165812 440791 662640 983001 723185 838099 256041 992095 488058 655225 713460 295500 498290 846717 615802 076650 106943 066009 675832 824160 > 4232 [i]