Best Known (55, 210, s)-Nets in Base 4
(55, 210, 66)-Net over F4 — Constructive and digital
Digital (55, 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
(55, 210, 91)-Net over F4 — Digital
Digital (55, 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
(55, 210, 277)-Net over F4 — Upper bound on s (digital)
There is no digital (55, 210, 278)-net over F4, because
- 3 times m-reduction [i] would yield digital (55, 207, 278)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4207, 278, F4, 152) (dual of [278, 71, 153]-code), but
- residual code [i] would yield OA(455, 125, S4, 38), but
- the linear programming bound shows that M ≥ 319 069835 724865 932418 074123 003651 341257 341197 442579 627763 661845 151026 883688 745199 999424 563055 993464 060641 280000 / 235791 539716 773839 952561 097991 148290 075568 550579 267395 341876 457603 747103 745171 > 455 [i]
- residual code [i] would yield OA(455, 125, S4, 38), but
- extracting embedded orthogonal array [i] would yield linear OA(4207, 278, F4, 152) (dual of [278, 71, 153]-code), but
(55, 210, 363)-Net in Base 4 — Upper bound on s
There is no (55, 210, 364)-net in base 4, because
- 1 times m-reduction [i] would yield (55, 209, 364)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 805200 278938 896038 795575 040908 101453 200720 304980 610592 697885 458886 179925 367636 778099 569697 715261 513626 433074 095480 511539 210280 > 4209 [i]