Best Known (174−32, 174, s)-Nets in Base 4
(174−32, 174, 1158)-Net over F4 — Constructive and digital
Digital (142, 174, 1158)-net over F4, using
- 42 times duplication [i] based on digital (140, 172, 1158)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (28, 44, 130)-net over F4, using
- trace code for nets [i] based on digital (6, 22, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 22, 65)-net over F16, using
- digital (96, 128, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 32, 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, 32, 257)-net over F256, using
- digital (28, 44, 130)-net over F4, using
- (u, u+v)-construction [i] based on
(174−32, 174, 11876)-Net over F4 — Digital
Digital (142, 174, 11876)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4174, 11876, F4, 32) (dual of [11876, 11702, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4174, 16410, F4, 32) (dual of [16410, 16236, 33]-code), using
- strength reduction [i] based on linear OA(4174, 16410, F4, 33) (dual of [16410, 16236, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- linear OA(4169, 16384, F4, 33) (dual of [16384, 16215, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- strength reduction [i] based on linear OA(4174, 16410, F4, 33) (dual of [16410, 16236, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4174, 16410, F4, 32) (dual of [16410, 16236, 33]-code), using
(174−32, 174, 7995013)-Net in Base 4 — Upper bound on s
There is no (142, 174, 7995014)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 573 375712 376070 476562 205907 290961 959659 329800 886489 466754 221382 645456 877302 614188 703111 815092 592256 159160 > 4174 [i]