Best Known (68−24, 68, s)-Nets in Base 16
(68−24, 68, 581)-Net over F16 — Constructive and digital
Digital (44, 68, 581)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 18, 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 (26, 50, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 25, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 25, 258)-net over F256, using
- digital (6, 18, 65)-net over F16, using
(68−24, 68, 594)-Net in Base 16 — Constructive
(44, 68, 594)-net in base 16, using
- (u, u+v)-construction [i] based on
- (8, 20, 80)-net in base 16, using
- 1 times m-reduction [i] based on (8, 21, 80)-net in base 16, using
- base change [i] based on digital (1, 14, 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, 14, 80)-net over F64, using
- 1 times m-reduction [i] based on (8, 21, 80)-net in base 16, using
- digital (24, 48, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- (8, 20, 80)-net in base 16, using
(68−24, 68, 2793)-Net over F16 — Digital
Digital (44, 68, 2793)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1668, 2793, F16, 24) (dual of [2793, 2725, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(1668, 4103, F16, 24) (dual of [4103, 4035, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(1667, 4096, F16, 24) (dual of [4096, 4029, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 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(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(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(1668, 4103, F16, 24) (dual of [4103, 4035, 25]-code), using
(68−24, 68, 2347553)-Net in Base 16 — Upper bound on s
There is no (44, 68, 2347554)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 7588 557378 634349 645172 525064 297576 539645 391873 121484 673989 685413 589262 246063 965096 > 1668 [i]