Best Known (154, 154+37, s)-Nets in Base 4
(154, 154+37, 1104)-Net over F4 — Constructive and digital
Digital (154, 191, 1104)-net over F4, using
- 41 times duplication [i] based on digital (153, 190, 1104)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (24, 42, 76)-net over F4, using
- trace code for nets [i] based on digital (3, 21, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- trace code for nets [i] based on digital (3, 21, 38)-net over F16, 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 (24, 42, 76)-net over F4, using
- (u, u+v)-construction [i] based on
(154, 154+37, 8572)-Net over F4 — Digital
Digital (154, 191, 8572)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4191, 8572, F4, 37) (dual of [8572, 8381, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 16392, F4, 37) (dual of [16392, 16201, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4190, 16384, F4, 37) (dual of [16384, 16194, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4183, 16384, F4, 35) (dual of [16384, 16201, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 16392, F4, 37) (dual of [16392, 16201, 38]-code), using
(154, 154+37, 5702763)-Net in Base 4 — Upper bound on s
There is no (154, 191, 5702764)-net in base 4, because
- 1 times m-reduction [i] would yield (154, 190, 5702764)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 462626 959302 544201 348660 479853 102288 422662 108299 919420 897876 064768 723220 462082 434096 092340 134666 331578 413262 048410 > 4190 [i]