Best Known (43−14, 43, s)-Nets in Base 16
(43−14, 43, 1028)-Net over F16 — Constructive and digital
Digital (29, 43, 1028)-net over F16, using
- 1 times m-reduction [i] based on digital (29, 44, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (7, 14, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 257)-net over F256, using
- digital (15, 30, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- digital (7, 14, 514)-net over F16, using
- (u, u+v)-construction [i] based on
(43−14, 43, 4240)-Net over F16 — Digital
Digital (29, 43, 4240)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1643, 4240, F16, 14) (dual of [4240, 4197, 15]-code), using
- 138 step Varšamov–Edel lengthening with (ri) = (2, 13 times 0, 1, 123 times 0) [i] based on linear OA(1640, 4099, F16, 14) (dual of [4099, 4059, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(1640, 4096, F16, 14) (dual of [4096, 4056, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(1637, 4096, F16, 13) (dual of [4096, 4059, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- 138 step Varšamov–Edel lengthening with (ri) = (2, 13 times 0, 1, 123 times 0) [i] based on linear OA(1640, 4099, F16, 14) (dual of [4099, 4059, 15]-code), using
(43−14, 43, 5617772)-Net in Base 16 — Upper bound on s
There is no (29, 43, 5617773)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 5986 313519 490781 658624 203275 797157 316354 491685 539016 > 1643 [i]