Best Known (137, 137+30, s)-Nets in Base 4
(137, 137+30, 1223)-Net over F4 — Constructive and digital
Digital (137, 167, 1223)-net over F4, using
- 42 times duplication [i] based on digital (135, 165, 1223)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (30, 45, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 15, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 15, 65)-net over F64, using
- digital (90, 120, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 30, 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, 30, 257)-net over F256, using
- digital (30, 45, 195)-net over F4, using
- (u, u+v)-construction [i] based on
(137, 137+30, 13949)-Net over F4 — Digital
Digital (137, 167, 13949)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4167, 13949, F4, 30) (dual of [13949, 13782, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4167, 16424, F4, 30) (dual of [16424, 16257, 31]-code), using
- 5 times code embedding in larger space [i] based on linear OA(4162, 16419, F4, 30) (dual of [16419, 16257, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(4155, 16384, F4, 30) (dual of [16384, 16229, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4127, 16384, F4, 25) (dual of [16384, 16257, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- 5 times code embedding in larger space [i] based on linear OA(4162, 16419, F4, 30) (dual of [16419, 16257, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4167, 16424, F4, 30) (dual of [16424, 16257, 31]-code), using
(137, 137+30, large)-Net in Base 4 — Upper bound on s
There is no (137, 167, large)-net in base 4, because
- 28 times m-reduction [i] would yield (137, 139, large)-net in base 4, but