Best Known (21−7, 21, s)-Nets in Base 4
(21−7, 21, 195)-Net over F4 — Constructive and digital
Digital (14, 21, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 7, 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
(21−7, 21, 219)-Net over F4 — Digital
Digital (14, 21, 219)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(421, 219, F4, 7) (dual of [219, 198, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(421, 255, F4, 7) (dual of [255, 234, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(421, 255, F4, 7) (dual of [255, 234, 8]-code), using
(21−7, 21, 6249)-Net in Base 4 — Upper bound on s
There is no (14, 21, 6250)-net in base 4, because
- 1 times m-reduction [i] would yield (14, 20, 6250)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 099687 696876 > 420 [i]