Best Known (44, 66, s)-Nets in Base 16
(44, 66, 1028)-Net over F16 — Constructive and digital
Digital (44, 66, 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 (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 (see above)
- trace code for nets [i] based on digital (0, 22, 257)-net over F256, using
- digital (11, 22, 514)-net over F16, using
(44, 66, 4182)-Net over F16 — Digital
Digital (44, 66, 4182)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1666, 4182, F16, 22) (dual of [4182, 4116, 23]-code), using
- 78 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 15 times 0, 1, 57 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
- 78 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 15 times 0, 1, 57 times 0) [i] based on linear OA(1661, 4099, F16, 22) (dual of [4099, 4038, 23]-code), using
(44, 66, 5490884)-Net in Base 16 — Upper bound on s
There is no (44, 66, 5490885)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 29 642781 176312 407825 132600 290590 901431 166089 547733 112017 498008 312887 512222 679776 > 1666 [i]