Best Known (65, 65+157, s)-Nets in Base 4
(65, 65+157, 66)-Net over F4 — Constructive and digital
Digital (65, 222, 66)-net over F4, using
- t-expansion [i] based on digital (49, 222, 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
(65, 65+157, 99)-Net over F4 — Digital
Digital (65, 222, 99)-net over F4, using
- t-expansion [i] based on digital (61, 222, 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
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
(65, 65+157, 433)-Net over F4 — Upper bound on s (digital)
There is no digital (65, 222, 434)-net over F4, because
- 1 times m-reduction [i] would yield digital (65, 221, 434)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4221, 434, F4, 156) (dual of [434, 213, 157]-code), but
- residual code [i] would yield OA(465, 277, S4, 39), but
- the linear programming bound shows that M ≥ 114860 870016 374579 218653 717961 476899 219304 784975 141224 743527 918706 993971 303087 603712 / 81 574206 852274 374488 234675 560995 174637 976327 > 465 [i]
- residual code [i] would yield OA(465, 277, S4, 39), but
- extracting embedded orthogonal array [i] would yield linear OA(4221, 434, F4, 156) (dual of [434, 213, 157]-code), but
(65, 65+157, 443)-Net in Base 4 — Upper bound on s
There is no (65, 222, 444)-net in base 4, because
- 1 times m-reduction [i] would yield (65, 221, 444)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 11 360605 748267 414006 716214 265251 091416 172483 640073 054366 781144 035522 594163 661138 521814 110156 754464 077791 374857 785933 175230 920101 519320 > 4221 [i]