Best Known (245−174, 245, s)-Nets in Base 4
(245−174, 245, 66)-Net over F4 — Constructive and digital
Digital (71, 245, 66)-net over F4, using
- t-expansion [i] based on digital (49, 245, 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
(245−174, 245, 105)-Net over F4 — Digital
Digital (71, 245, 105)-net over F4, using
- t-expansion [i] based on digital (70, 245, 105)-net over F4, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 70 and N(F) ≥ 105, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
(245−174, 245, 453)-Net over F4 — Upper bound on s (digital)
There is no digital (71, 245, 454)-net over F4, because
- 2 times m-reduction [i] would yield digital (71, 243, 454)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4243, 454, F4, 172) (dual of [454, 211, 173]-code), but
- residual code [i] would yield OA(471, 281, S4, 43), but
- the linear programming bound shows that M ≥ 373 183646 729416 144099 480364 087718 649481 545879 174441 862191 129687 019685 715860 929497 289195 520000 / 66 442053 478467 341276 388448 356669 192467 605369 216717 > 471 [i]
- residual code [i] would yield OA(471, 281, S4, 43), but
- extracting embedded orthogonal array [i] would yield linear OA(4243, 454, F4, 172) (dual of [454, 211, 173]-code), but
(245−174, 245, 480)-Net in Base 4 — Upper bound on s
There is no (71, 245, 481)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3732 618276 388640 072891 390463 460252 520683 786683 378365 430818 150526 985302 744103 317063 380512 779201 975289 799902 354789 897726 632601 474988 910295 144349 284000 > 4245 [i]