Best Known (223−162, 223, s)-Nets in Base 4
(223−162, 223, 66)-Net over F4 — Constructive and digital
Digital (61, 223, 66)-net over F4, using
- t-expansion [i] based on digital (49, 223, 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
(223−162, 223, 99)-Net over F4 — Digital
Digital (61, 223, 99)-net over F4, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 61 and N(F) ≥ 99, using
(223−162, 223, 335)-Net over F4 — Upper bound on s (digital)
There is no digital (61, 223, 336)-net over F4, because
- 2 times m-reduction [i] would yield digital (61, 221, 336)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4221, 336, F4, 160) (dual of [336, 115, 161]-code), but
- residual code [i] would yield OA(461, 175, S4, 40), but
- the linear programming bound shows that M ≥ 1 798315 977868 370852 713534 848248 812726 194497 404051 298068 679657 800761 225918 150102 236050 489344 / 313133 764813 349339 900431 732581 184165 925845 010666 409433 > 461 [i]
- residual code [i] would yield OA(461, 175, S4, 40), but
- extracting embedded orthogonal array [i] would yield linear OA(4221, 336, F4, 160) (dual of [336, 115, 161]-code), but
(223−162, 223, 405)-Net in Base 4 — Upper bound on s
There is no (61, 223, 406)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 196 733340 494610 062804 798175 238976 943363 358559 837542 517706 780103 665369 257385 376926 321154 447717 421116 719384 108881 396148 968037 346108 615414 > 4223 [i]