Best Known (49, 49+24, s)-Nets in Base 16
(49, 49+24, 1028)-Net over F16 — Constructive and digital
Digital (49, 73, 1028)-net over F16, using
- 1 times m-reduction [i] based on digital (49, 74, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (12, 24, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- digital (25, 50, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 25, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 25, 257)-net over F256, using
- digital (12, 24, 514)-net over F16, using
- (u, u+v)-construction [i] based on
(49, 49+24, 4375)-Net over F16 — Digital
Digital (49, 73, 4375)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1673, 4375, F16, 24) (dual of [4375, 4302, 25]-code), using
- 270 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 16 times 0, 1, 61 times 0, 1, 186 times 0) [i] based on linear OA(1667, 4099, F16, 24) (dual of [4099, 4032, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [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(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(160, 3, F16, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- 270 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 16 times 0, 1, 61 times 0, 1, 186 times 0) [i] based on linear OA(1667, 4099, F16, 24) (dual of [4099, 4032, 25]-code), using
(49, 49+24, 7453032)-Net in Base 16 — Upper bound on s
There is no (49, 73, 7453033)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 7957 178122 757073 158889 022857 265959 717553 193215 568766 672002 835277 863524 607108 808572 767416 > 1673 [i]