Best Known (45−16, 45, s)-Nets in Base 16
(45−16, 45, 563)-Net over F16 — Constructive and digital
Digital (29, 45, 563)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (5, 13, 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 (16, 32, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 16, 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, 16, 257)-net over F256, using
- digital (5, 13, 49)-net over F16, using
(45−16, 45, 579)-Net in Base 16 — Constructive
(29, 45, 579)-net in base 16, using
- 1 times m-reduction [i] based on (29, 46, 579)-net in base 16, using
- (u, u+v)-construction [i] based on
- (4, 12, 65)-net in base 16, using
- base change [i] based on digital (0, 8, 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, 8, 65)-net over F64, using
- digital (17, 34, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 17, 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, 17, 257)-net over F256, using
- (4, 12, 65)-net in base 16, using
- (u, u+v)-construction [i] based on
(45−16, 45, 2446)-Net over F16 — Digital
Digital (29, 45, 2446)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1645, 2446, F16, 16) (dual of [2446, 2401, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(1645, 4095, F16, 16) (dual of [4095, 4050, 17]-code), using
- 1 times truncation [i] based on linear OA(1646, 4096, F16, 17) (dual of [4096, 4050, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 1 times truncation [i] based on linear OA(1646, 4096, F16, 17) (dual of [4096, 4050, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1645, 4095, F16, 16) (dual of [4095, 4050, 17]-code), using
(45−16, 45, 1488581)-Net in Base 16 — Upper bound on s
There is no (29, 45, 1488582)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 1 532502 064463 793269 068791 225546 967611 214873 232381 466616 > 1645 [i]