Best Known (19, 19+9, s)-Nets in Base 4
(19, 19+9, 195)-Net over F4 — Constructive and digital
Digital (19, 28, 195)-net over F4, using
- 41 times duplication [i] based on digital (18, 27, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 9, 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, 9, 65)-net over F64, using
(19, 19+9, 232)-Net over F4 — Digital
Digital (19, 28, 232)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(428, 232, F4, 9) (dual of [232, 204, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(428, 255, F4, 9) (dual of [255, 227, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(428, 255, F4, 9) (dual of [255, 227, 10]-code), using
(19, 19+9, 8544)-Net in Base 4 — Upper bound on s
There is no (19, 28, 8545)-net in base 4, because
- 1 times m-reduction [i] would yield (19, 27, 8545)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 18017 639013 753256 > 427 [i]