Best Known (181−37, 181, s)-Nets in Base 4
(181−37, 181, 1061)-Net over F4 — Constructive and digital
Digital (144, 181, 1061)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (15, 33, 33)-net over F4, using
- net from sequence [i] based on digital (15, 32)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 15 and N(F) ≥ 33, using
- net from sequence [i] based on digital (15, 32)-sequence over F4, using
- digital (111, 148, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 37, 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, 37, 257)-net over F256, using
- digital (15, 33, 33)-net over F4, using
(181−37, 181, 5086)-Net over F4 — Digital
Digital (144, 181, 5086)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4181, 5086, F4, 37) (dual of [5086, 4905, 38]-code), using
- 972 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 1, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 14 times 0, 1, 22 times 0, 1, 35 times 0, 1, 52 times 0, 1, 75 times 0, 1, 102 times 0, 1, 130 times 0, 1, 154 times 0, 1, 172 times 0, 1, 185 times 0) [i] based on linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 972 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 1, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 14 times 0, 1, 22 times 0, 1, 35 times 0, 1, 52 times 0, 1, 75 times 0, 1, 102 times 0, 1, 130 times 0, 1, 154 times 0, 1, 172 times 0, 1, 185 times 0) [i] based on linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using
(181−37, 181, 2640014)-Net in Base 4 — Upper bound on s
There is no (144, 181, 2640015)-net in base 4, because
- 1 times m-reduction [i] would yield (144, 180, 2640015)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 348548 861956 266109 503057 163526 132290 074260 650654 487318 728312 225992 436051 327702 167668 162673 042230 555842 198040 > 4180 [i]