Best Known (55−20, 55, s)-Nets in Base 16
(55−20, 55, 563)-Net over F16 — Constructive and digital
Digital (35, 55, 563)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (5, 15, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- digital (20, 40, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 20, 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, 20, 257)-net over F256, using
- digital (5, 15, 49)-net over F16, using
(55−20, 55, 579)-Net in Base 16 — Constructive
(35, 55, 579)-net in base 16, using
- (u, u+v)-construction [i] based on
- (5, 15, 65)-net in base 16, using
- base change [i] based on digital (0, 10, 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
- base change [i] based on digital (0, 10, 65)-net over F64, using
- digital (20, 40, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 20, 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, 20, 257)-net over F256, using
- (5, 15, 65)-net in base 16, using
(55−20, 55, 2053)-Net over F16 — Digital
Digital (35, 55, 2053)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1655, 2053, F16, 20) (dual of [2053, 1998, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1655, 4096, F16, 20) (dual of [4096, 4041, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(1655, 4096, F16, 20) (dual of [4096, 4041, 21]-code), using
(55−20, 55, 1266319)-Net in Base 16 — Upper bound on s
There is no (35, 55, 1266320)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 1 685007 886901 418681 884263 916735 625823 330147 370168 133546 817969 274251 > 1655 [i]