Best Known (42, 65, s)-Nets in Base 16
(42, 65, 581)-Net over F16 — Constructive and digital
Digital (42, 65, 581)-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 (25, 48, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 24, 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, 24, 258)-net over F256, using
- digital (6, 17, 65)-net over F16, using
(42, 65, 594)-Net in Base 16 — Constructive
(42, 65, 594)-net in base 16, using
- 161 times duplication [i] based on (41, 64, 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 (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
- (7, 18, 80)-net in base 16, using
- (u, u+v)-construction [i] based on
(42, 65, 2694)-Net over F16 — Digital
Digital (42, 65, 2694)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1665, 2694, F16, 23) (dual of [2694, 2629, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(1665, 4103, F16, 23) (dual of [4103, 4038, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(1664, 4096, F16, 23) (dual of [4096, 4032, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(1658, 4096, F16, 21) (dual of [4096, 4038, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(1665, 4103, F16, 23) (dual of [4103, 4038, 24]-code), using
(42, 65, 3316737)-Net in Base 16 — Upper bound on s
There is no (42, 65, 3316738)-net in base 16, because
- 1 times m-reduction [i] would yield (42, 64, 3316738)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 115792 177592 419573 252905 991277 118075 018117 985890 531770 028197 769718 118634 512896 > 1664 [i]