Best Known (45, 67, s)-Nets in Base 16
(45, 67, 1028)-Net over F16 — Constructive and digital
Digital (45, 67, 1028)-net over F16, using
- 1 times m-reduction [i] based on digital (45, 68, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (11, 22, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 257)-net over F256, 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 (see above)
- trace code for nets [i] based on digital (0, 23, 257)-net over F256, using
- digital (11, 22, 514)-net over F16, using
- (u, u+v)-construction [i] based on
(45, 67, 4350)-Net over F16 — Digital
Digital (45, 67, 4350)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1667, 4350, F16, 22) (dual of [4350, 4283, 23]-code), using
- 245 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 15 times 0, 1, 57 times 0, 1, 166 times 0) [i] based on linear OA(1661, 4099, F16, 22) (dual of [4099, 4038, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [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(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(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(21) ⊂ Ce(20) [i] based on
- 245 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 15 times 0, 1, 57 times 0, 1, 166 times 0) [i] based on linear OA(1661, 4099, F16, 22) (dual of [4099, 4038, 23]-code), using
(45, 67, 7064930)-Net in Base 16 — Upper bound on s
There is no (45, 67, 7064931)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 474 284770 534347 134351 261011 553087 457240 326859 922650 970359 560847 029749 461454 611616 > 1667 [i]