Best Known (229−170, 229, s)-Nets in Base 4
(229−170, 229, 66)-Net over F4 — Constructive and digital
Digital (59, 229, 66)-net over F4, using
- t-expansion [i] based on digital (49, 229, 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
(229−170, 229, 91)-Net over F4 — Digital
Digital (59, 229, 91)-net over F4, using
- t-expansion [i] based on digital (50, 229, 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
(229−170, 229, 271)-Net over F4 — Upper bound on s (digital)
There is no digital (59, 229, 272)-net over F4, because
- 2 times m-reduction [i] would yield digital (59, 227, 272)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4227, 272, F4, 168) (dual of [272, 45, 169]-code), but
- residual code [i] would yield OA(459, 103, S4, 42), but
- the linear programming bound shows that M ≥ 20 209278 630007 954817 311657 592716 406577 471235 077989 401051 636395 089582 131646 221319 183990 947510 747136 / 59 997456 042944 145499 206694 218235 767252 553003 337898 939171 700875 > 459 [i]
- residual code [i] would yield OA(459, 103, S4, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(4227, 272, F4, 168) (dual of [272, 45, 169]-code), but
(229−170, 229, 386)-Net in Base 4 — Upper bound on s
There is no (59, 229, 387)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 818493 655002 078853 376103 886399 680184 797871 950791 974684 214277 786623 061710 222311 343518 321208 016181 689198 886510 523288 803168 182720 143111 707072 > 4229 [i]