Best Known (208−153, 208, s)-Nets in Base 4
(208−153, 208, 66)-Net over F4 — Constructive and digital
Digital (55, 208, 66)-net over F4, using
- t-expansion [i] based on digital (49, 208, 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
(208−153, 208, 91)-Net over F4 — Digital
Digital (55, 208, 91)-net over F4, using
- t-expansion [i] based on digital (50, 208, 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
(208−153, 208, 277)-Net over F4 — Upper bound on s (digital)
There is no digital (55, 208, 278)-net over F4, because
- 1 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
(208−153, 208, 363)-Net in Base 4 — Upper bound on s
There is no (55, 208, 364)-net in base 4, because
- 1 times m-reduction [i] would yield (55, 207, 364)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 42550 759292 619671 374497 695497 504229 567560 908985 395151 952514 238244 200354 031721 481063 042944 341633 132481 455361 756144 321210 991288 > 4207 [i]