Best Known (162−33, 162, s)-Nets in Base 4
(162−33, 162, 1060)-Net over F4 — Constructive and digital
Digital (129, 162, 1060)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (14, 30, 32)-net over F4, using
- 2 times m-reduction [i] based on digital (14, 32, 32)-net over F4, using
- digital (99, 132, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 33, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 33, 257)-net over F256, using
- digital (14, 30, 32)-net over F4, using
(162−33, 162, 4811)-Net over F4 — Digital
Digital (129, 162, 4811)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4162, 4811, F4, 33) (dual of [4811, 4649, 34]-code), using
- 697 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 12 times 0, 1, 18 times 0, 1, 29 times 0, 1, 43 times 0, 1, 62 times 0, 1, 85 times 0, 1, 113 times 0, 1, 141 times 0, 1, 168 times 0) [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 697 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 12 times 0, 1, 18 times 0, 1, 29 times 0, 1, 43 times 0, 1, 62 times 0, 1, 85 times 0, 1, 113 times 0, 1, 141 times 0, 1, 168 times 0) [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
(162−33, 162, 2592053)-Net in Base 4 — Upper bound on s
There is no (129, 162, 2592054)-net in base 4, because
- 1 times m-reduction [i] would yield (129, 161, 2592054)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 8 543967 807986 833247 440480 134919 736201 212842 840156 629421 776596 136843 661152 681863 520037 757944 748867 > 4161 [i]