Best Known (62−22, 62, s)-Nets in Base 16
(62−22, 62, 579)-Net over F16 — Constructive and digital
Digital (40, 62, 579)-net over F16, using
- 1 times m-reduction [i] based on digital (40, 63, 579)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 17, 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
- digital (23, 46, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 23, 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, 23, 257)-net over F256, using
- digital (6, 17, 65)-net over F16, using
- (u, u+v)-construction [i] based on
(62−22, 62, 594)-Net in Base 16 — Constructive
(40, 62, 594)-net in base 16, using
- (u, u+v)-construction [i] based on
- (7, 18, 80)-net in base 16, using
- base change [i] based on digital (1, 12, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- base change [i] based on digital (1, 12, 80)-net over F64, using
- digital (22, 44, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 22, 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, 22, 257)-net over F256, using
- (7, 18, 80)-net in base 16, using
(62−22, 62, 2595)-Net over F16 — Digital
Digital (40, 62, 2595)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1662, 2595, F16, 22) (dual of [2595, 2533, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(1662, 4103, F16, 22) (dual of [4103, 4041, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(1661, 4096, F16, 22) (dual of [4096, 4035, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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]
- linear OA(161, 7, F16, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(1662, 4103, F16, 22) (dual of [4103, 4041, 23]-code), using
(62−22, 62, 2003455)-Net in Base 16 — Upper bound on s
There is no (40, 62, 2003456)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 452 313656 415358 193835 161360 488496 378429 856596 239483 254400 430527 556484 018241 > 1662 [i]